numerical methods - oridnary differential equations - 2
TRANSCRIPT
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Numerical MethodsOrdinary Differential Equations - 2
Dr. N. B. Vyas
Department of Mathematics,Atmiya Institute of Tech. and Science,
Rajkot (Gujarat) - [email protected]
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Ordinary Differential Equations
Euler’s Method:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
The Taylor’s series is
y(x) = y(x0) +(x− x0)
1!y′(x0) +
(x− x0)2
2!y′′(x0) + . . . - - - (1)
Now substituting h = x1 − x0 in eq (1), we get
y(x1) = y(x0) + hy′(x0) +h2
2!y′′(x0) + . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 3: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/3.jpg)
Ordinary Differential Equations
Euler’s Method:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
The Taylor’s series is
y(x) = y(x0) +(x− x0)
1!y′(x0) +
(x− x0)2
2!y′′(x0) + . . . - - - (1)
Now substituting h = x1 − x0 in eq (1), we get
y(x1) = y(x0) + hy′(x0) +h2
2!y′′(x0) + . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 4: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/4.jpg)
Ordinary Differential Equations
Euler’s Method:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
The Taylor’s series is
y(x) = y(x0) +(x− x0)
1!y′(x0) +
(x− x0)2
2!y′′(x0) + . . . - - - (1)
Now substituting h = x1 − x0 in eq (1), we get
y(x1) = y(x0) + hy′(x0) +h2
2!y′′(x0) + . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 5: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/5.jpg)
Ordinary Differential Equations
Euler’s Method:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
The Taylor’s series is
y(x) = y(x0) +(x− x0)
1!y′(x0) +
(x− x0)2
2!y′′(x0) + . . . - - - (1)
Now substituting h = x1 − x0 in eq (1), we get
y(x1) = y(x0) + hy′(x0) +h2
2!y′′(x0) + . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 6: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/6.jpg)
Euler’s Method
If h is chosen small enough then we may neglect the second andhigher order term of h.
y1 = y0 + hf(x0, y0)Which is Euler’s first approximation.The general step for Euler method isyi+1 = yi + hf(xi, yi) where i = 0, 1, 2....
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
If h is chosen small enough then we may neglect the second andhigher order term of h.y1 = y0 + hf(x0, y0)
Which is Euler’s first approximation.The general step for Euler method isyi+1 = yi + hf(xi, yi) where i = 0, 1, 2....
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
If h is chosen small enough then we may neglect the second andhigher order term of h.y1 = y0 + hf(x0, y0)Which is Euler’s first approximation.
The general step for Euler method isyi+1 = yi + hf(xi, yi) where i = 0, 1, 2....
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 9: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/9.jpg)
Euler’s Method
If h is chosen small enough then we may neglect the second andhigher order term of h.y1 = y0 + hf(x0, y0)Which is Euler’s first approximation.The general step for Euler method is
yi+1 = yi + hf(xi, yi) where i = 0, 1, 2....
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 10: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/10.jpg)
Euler’s Method
If h is chosen small enough then we may neglect the second andhigher order term of h.y1 = y0 + hf(x0, y0)Which is Euler’s first approximation.The general step for Euler method isyi+1 = yi + hf(xi, yi) where i = 0, 1, 2....
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 11: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/11.jpg)
Euler’s Method
Ex.: Use Euler’s method to find y(1.6) given thatdy
dx= xy
12 , y(1) = 1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)12
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
∴ y(1.6) = 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)12
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
∴ y(1.6) = 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 20: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/20.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)12
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
∴ y(1.6) = 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)12
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
∴ y(1.6) = 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 22: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/22.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 1.4629
x2 = x1 + h =
1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)12
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
∴ y(1.6) = 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 23: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/23.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)12
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
∴ y(1.6) = 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 24: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/24.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)12
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
∴ y(1.6) = 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 25: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/25.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)12
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
∴ y(1.6) = 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 26: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/26.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)12
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
∴ y(1.6) = 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 27: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/27.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)12
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
∴ y(1.6) = 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Ex.: Using Euler’s method, find y(0.2), givendy
dx= y − 2x
y, y(0) = 1. (Take h = 0.1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
y
we take h = 0.1
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(1 − 0)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe take h = 0.1
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(1 − 0)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe take h = 0.1
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(1 − 0)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe take h = 0.1
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(1 − 0)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 33: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/33.jpg)
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe take h = 0.1
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(1 − 0)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 34: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/34.jpg)
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe take h = 0.1
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(1 − 0)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe take h = 0.1
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(1 − 0)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)
(0.1 − 2(0.1)
1.1
)= 1.1918
x2 = x1 + h = 0.1 + 0.1 = 0.2
∴ y(0.2) = 1.1918
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)
(0.1 − 2(0.1)
1.1
)= 1.1918
x2 = x1 + h = 0.1 + 0.1 = 0.2
∴ y(0.2) = 1.1918
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)
(0.1 − 2(0.1)
1.1
)= 1.1918
x2 = x1 + h = 0.1 + 0.1 = 0.2
∴ y(0.2) = 1.1918
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)
(0.1 − 2(0.1)
1.1
)
= 1.1918
x2 = x1 + h = 0.1 + 0.1 = 0.2
∴ y(0.2) = 1.1918
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)
(0.1 − 2(0.1)
1.1
)= 1.1918
x2 = x1 + h = 0.1 + 0.1 = 0.2
∴ y(0.2) = 1.1918
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 41: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/41.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)
(0.1 − 2(0.1)
1.1
)= 1.1918
x2 = x1 + h = 0.1 + 0.1 = 0.2
∴ y(0.2) = 1.1918
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 42: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/42.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)
(0.1 − 2(0.1)
1.1
)= 1.1918
x2 = x1 + h = 0.1 + 0.1 = 0.2
∴ y(0.2) = 1.1918
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 43: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/43.jpg)
Euler’s Method
Ex.: Use Euler’s method to obtain an approx value
of y(0.4) for the equationdy
dx= x + y, y(0) = 1 with
h = 0.1.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
dx= f(x, y) = x + y
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(0 + 1)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 45: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/45.jpg)
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
dx= f(x, y) = x + y
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(0 + 1)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 46: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/46.jpg)
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
dx= f(x, y) = x + y
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(0 + 1)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 47: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/47.jpg)
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
dx= f(x, y) = x + y
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(0 + 1)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 48: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/48.jpg)
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
dx= f(x, y) = x + y
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(0 + 1)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 49: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/49.jpg)
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
dx= f(x, y) = x + y
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(0 + 1)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 50: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/50.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 51: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/51.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 52: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/52.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 53: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/53.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 54: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/54.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 55: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/55.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 56: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/56.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 57: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/57.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 58: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/58.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 59: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/59.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 60: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/60.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 61: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/61.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 62: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/62.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
= 1.362 + (0.1)(0.3 + 1.362)
= 1.5282
x4 = x3 + h = 0.3 + 0.1 = 0.4
∴ y(0.4) = 1.5282
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
= 1.362 + (0.1)(0.3 + 1.362)
= 1.5282
x4 = x3 + h = 0.3 + 0.1 = 0.4
∴ y(0.4) = 1.5282
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 64: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/64.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
= 1.362 + (0.1)(0.3 + 1.362)
= 1.5282
x4 = x3 + h = 0.3 + 0.1 = 0.4
∴ y(0.4) = 1.5282
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 65: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/65.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
= 1.362 + (0.1)(0.3 + 1.362)
= 1.5282
x4 = x3 + h = 0.3 + 0.1 = 0.4
∴ y(0.4) = 1.5282
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 66: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/66.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
= 1.362 + (0.1)(0.3 + 1.362)
= 1.5282
x4 = x3 + h = 0.3 + 0.1 = 0.4
∴ y(0.4) = 1.5282
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 67: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/67.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
= 1.362 + (0.1)(0.3 + 1.362)
= 1.5282
x4 = x3 + h = 0.3 + 0.1 = 0.4
∴ y(0.4) = 1.5282
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 68: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/68.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
= 1.362 + (0.1)(0.3 + 1.362)
= 1.5282
x4 = x3 + h = 0.3 + 0.1 = 0.4
∴ y(0.4) = 1.5282
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Ex.: Givendy
dx=
y − x
y + x, y(0) = 1.
Find y(0.1) by Euler’s method in 5 steps.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
dx= f(x, y) =
y − x
y + x
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.02)
(1 − 0
1 + 0
)= 1.02
x1 = x0 + h = 0 + 0.02 = 0.02
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
dx= f(x, y) =
y − x
y + x
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.02)
(1 − 0
1 + 0
)= 1.02
x1 = x0 + h = 0 + 0.02 = 0.02
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
dx= f(x, y) =
y − x
y + x
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.02)
(1 − 0
1 + 0
)= 1.02
x1 = x0 + h = 0 + 0.02 = 0.02
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
dx= f(x, y) =
y − x
y + x
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.02)
(1 − 0
1 + 0
)
= 1.02
x1 = x0 + h = 0 + 0.02 = 0.02
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
dx= f(x, y) =
y − x
y + x
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.02)
(1 − 0
1 + 0
)= 1.02
x1 = x0 + h = 0 + 0.02 = 0.02
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
dx= f(x, y) =
y − x
y + x
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.02)
(1 − 0
1 + 0
)= 1.02
x1 = x0 + h = 0 + 0.02 = 0.02
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.02 + (0.02)
(1.02 − 0.02
1.02 + 0.02
)= 1.0392
x2 = x1 + h = 0.02 + 0.02 = 0.04
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.02 + (0.02)
(1.02 − 0.02
1.02 + 0.02
)= 1.0392
x2 = x1 + h = 0.02 + 0.02 = 0.04
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 78: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/78.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.02 + (0.02)
(1.02 − 0.02
1.02 + 0.02
)
= 1.0392
x2 = x1 + h = 0.02 + 0.02 = 0.04
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 79: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/79.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.02 + (0.02)
(1.02 − 0.02
1.02 + 0.02
)= 1.0392
x2 = x1 + h = 0.02 + 0.02 = 0.04
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 80: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/80.jpg)
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.02 + (0.02)
(1.02 − 0.02
1.02 + 0.02
)= 1.0392
x2 = x1 + h = 0.02 + 0.02 = 0.04
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 81: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/81.jpg)
Euler’s Method
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.0392 + (0.02)
(1.0392 − 0.04
1.0392 + 0.04
)= 1.0577
x3 = x2 + h = 0.04 + 0.02 = 0.06
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 82: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/82.jpg)
Euler’s Method
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.0392 + (0.02)
(1.0392 − 0.04
1.0392 + 0.04
)
= 1.0577
x3 = x2 + h = 0.04 + 0.02 = 0.06
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 83: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/83.jpg)
Euler’s Method
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.0392 + (0.02)
(1.0392 − 0.04
1.0392 + 0.04
)= 1.0577
x3 = x2 + h = 0.04 + 0.02 = 0.06
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 84: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/84.jpg)
Euler’s Method
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.0392 + (0.02)
(1.0392 − 0.04
1.0392 + 0.04
)= 1.0577
x3 = x2 + h = 0.04 + 0.02 = 0.06
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 85: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/85.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.0577 + (0.02)
(1.0577 − 0.06
1.0577 + 0.06
)= 1.0755
x4 = x3 + h = 0.06 + 0.02 = 0.08
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 86: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/86.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.0577 + (0.02)
(1.0577 − 0.06
1.0577 + 0.06
)= 1.0755
x4 = x3 + h = 0.06 + 0.02 = 0.08
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 87: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/87.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.0577 + (0.02)
(1.0577 − 0.06
1.0577 + 0.06
)
= 1.0755
x4 = x3 + h = 0.06 + 0.02 = 0.08
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 88: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/88.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.0577 + (0.02)
(1.0577 − 0.06
1.0577 + 0.06
)= 1.0755
x4 = x3 + h = 0.06 + 0.02 = 0.08
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 89: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/89.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.0577 + (0.02)
(1.0577 − 0.06
1.0577 + 0.06
)= 1.0755
x4 = x3 + h = 0.06 + 0.02 = 0.08
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 90: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/90.jpg)
Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.0577 + (0.02)
(1.0577 − 0.06
1.0577 + 0.06
)= 1.0755
x4 = x3 + h = 0.06 + 0.02 = 0.08
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 91: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/91.jpg)
Euler’s Method
5th approximation:
y4 = y3 + hf(x3, y3)
= 1.0755 + (0.02)
(1.0755 − 0.08
1.0755 + 0.08
)= 1.0928
x5 = x4 + h = 0.08 + 0.02 = 1
∴ y(1) = 1.0928
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 92: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/92.jpg)
Euler’s Method
5th approximation:
y4 = y3 + hf(x3, y3)
= 1.0755 + (0.02)
(1.0755 − 0.08
1.0755 + 0.08
)= 1.0928
x5 = x4 + h = 0.08 + 0.02 = 1
∴ y(1) = 1.0928
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 93: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/93.jpg)
Euler’s Method
5th approximation:
y4 = y3 + hf(x3, y3)
= 1.0755 + (0.02)
(1.0755 − 0.08
1.0755 + 0.08
)
= 1.0928
x5 = x4 + h = 0.08 + 0.02 = 1
∴ y(1) = 1.0928
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 94: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/94.jpg)
Euler’s Method
5th approximation:
y4 = y3 + hf(x3, y3)
= 1.0755 + (0.02)
(1.0755 − 0.08
1.0755 + 0.08
)= 1.0928
x5 = x4 + h = 0.08 + 0.02 = 1
∴ y(1) = 1.0928
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 95: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/95.jpg)
Euler’s Method
5th approximation:
y4 = y3 + hf(x3, y3)
= 1.0755 + (0.02)
(1.0755 − 0.08
1.0755 + 0.08
)= 1.0928
x5 = x4 + h = 0.08 + 0.02 = 1
∴ y(1) = 1.0928
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 96: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/96.jpg)
Euler’s Method
5th approximation:
y4 = y3 + hf(x3, y3)
= 1.0755 + (0.02)
(1.0755 − 0.08
1.0755 + 0.08
)= 1.0928
x5 = x4 + h = 0.08 + 0.02 = 1
∴ y(1) = 1.0928
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 97: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/97.jpg)
Euler’s Method
Ex.: Find y(2) fordy
dx=
y
x, y(1) = 1.
using Euler’s method, take h = 0.2.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y(1)1 of y1, we take
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
where x1 = x0 + h
For still better approximation y(2)1 of y1,
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
we repeat this process till two consecutive valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 99: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/99.jpg)
Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y(1)1 of y1, we take
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
where x1 = x0 + h
For still better approximation y(2)1 of y1,
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
we repeat this process till two consecutive valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 100: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/100.jpg)
Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y(1)1 of y1, we take
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
where x1 = x0 + h
For still better approximation y(2)1 of y1,
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
we repeat this process till two consecutive valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 101: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/101.jpg)
Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y(1)1 of y1, we take
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
where x1 = x0 + h
For still better approximation y(2)1 of y1,
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
we repeat this process till two consecutive valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 102: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/102.jpg)
Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y(1)1 of y1, we take
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
where x1 = x0 + h
For still better approximation y(2)1 of y1,
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
we repeat this process till two consecutive valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 103: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/103.jpg)
Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y(1)1 of y1, we take
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
where x1 = x0 + h
For still better approximation y(2)1 of y1,
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
we repeat this process till two consecutive valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 104: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/104.jpg)
Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y(1)1 of y1, we take
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
where x1 = x0 + h
For still better approximation y(2)1 of y1,
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
we repeat this process till two consecutive valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 105: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/105.jpg)
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2
y2 = y1 + hf(x1, y1)
For better approximation y(1)2 of y2, we take
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
where x2 = x1 + h
For still better approximation y(2)2 of y2,
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(2)2 )]
we repeat this step until y2 becomes stationary.Then we proceed to calculate y3 in the same wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 106: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/106.jpg)
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = y1 + hf(x1, y1)
For better approximation y(1)2 of y2, we take
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
where x2 = x1 + h
For still better approximation y(2)2 of y2,
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(2)2 )]
we repeat this step until y2 becomes stationary.Then we proceed to calculate y3 in the same wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 107: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/107.jpg)
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = y1 + hf(x1, y1)
For better approximation y(1)2 of y2, we take
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
where x2 = x1 + h
For still better approximation y(2)2 of y2,
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(2)2 )]
we repeat this step until y2 becomes stationary.Then we proceed to calculate y3 in the same wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 108: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/108.jpg)
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = y1 + hf(x1, y1)
For better approximation y(1)2 of y2, we take
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
where x2 = x1 + h
For still better approximation y(2)2 of y2,
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(2)2 )]
we repeat this step until y2 becomes stationary.Then we proceed to calculate y3 in the same wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 109: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/109.jpg)
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = y1 + hf(x1, y1)
For better approximation y(1)2 of y2, we take
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
where x2 = x1 + h
For still better approximation y(2)2 of y2,
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(2)2 )]
we repeat this step until y2 becomes stationary.Then we proceed to calculate y3 in the same wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 110: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/110.jpg)
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = y1 + hf(x1, y1)
For better approximation y(1)2 of y2, we take
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
where x2 = x1 + h
For still better approximation y(2)2 of y2,
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(2)2 )]
we repeat this step until y2 becomes stationary.Then we proceed to calculate y3 in the same wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 111: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/111.jpg)
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = y1 + hf(x1, y1)
For better approximation y(1)2 of y2, we take
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
where x2 = x1 + h
For still better approximation y(2)2 of y2,
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(2)2 )]
we repeat this step until y2 becomes stationary.Then we proceed to calculate y3 in the same wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 112: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/112.jpg)
Modified Euler’s Method
Ex.: Solvedy
dx= x + y , y(0) = 1.
by Euler’s modified method for x = 0.1
correct upto four decimal places by taking h = 0.05.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 anddy
dx= f(x, y) = x + y
x1 = x0 + h = 0 + 0.05 = 0.05
y1 = y0 + hf(x0, y0)
= 1 + (0.05)(1)
= 1.05
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 anddy
dx= f(x, y) = x + y
x1 = x0 + h = 0 + 0.05 = 0.05
y1 = y0 + hf(x0, y0)
= 1 + (0.05)(1)
= 1.05
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 115: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/115.jpg)
Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 anddy
dx= f(x, y) = x + y
x1 = x0 + h = 0 + 0.05 = 0.05
y1 = y0 + hf(x0, y0)
= 1 + (0.05)(1)
= 1.05
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 116: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/116.jpg)
Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 anddy
dx= f(x, y) = x + y
x1 = x0 + h = 0 + 0.05 = 0.05
y1 = y0 + hf(x0, y0)
= 1 + (0.05)(1)
= 1.05
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 anddy
dx= f(x, y) = x + y
x1 = x0 + h = 0 + 0.05 = 0.05
y1 = y0 + hf(x0, y0)
= 1 + (0.05)(1)
= 1.05
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.0525)] = 1.05256
∴ y1 = 1.05256 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 119: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/119.jpg)
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.0525)] = 1.05256
∴ y1 = 1.05256 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 120: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/120.jpg)
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.05)] =
1.0525
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.0525)] = 1.05256
∴ y1 = 1.05256 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.0525)] = 1.05256
∴ y1 = 1.05256 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 122: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/122.jpg)
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) +
f(x1, y(1)1 )]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.0525)] = 1.05256
∴ y1 = 1.05256 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 123: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/123.jpg)
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.0525)] = 1.05256
∴ y1 = 1.05256 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 124: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/124.jpg)
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.0525)] =
1.05256
∴ y1 = 1.05256 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 125: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/125.jpg)
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 1 + 0.052 [(0 + 1) + (0.05 + 1.0525)] = 1.05256
∴ y1 = 1.05256 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
x2 = x1 + h = 0.05 + 0.05 = 0.1
y2 = y1 + hf(x1, y1)
= 1.05256 + (0.05)(0.1 + 1.05256)
= 1.10769
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
x2 = x1 + h = 0.05 + 0.05 = 0.1
y2 = y1 + hf(x1, y1)
= 1.05256 + (0.05)(0.1 + 1.05256)
= 1.10769
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
x2 = x1 + h = 0.05 + 0.05 = 0.1
y2 = y1 + hf(x1, y1)
= 1.05256 + (0.05)(0.1 + 1.05256)
= 1.10769
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
x2 = x1 + h = 0.05 + 0.05 = 0.1
y2 = y1 + hf(x1, y1)
= 1.05256 + (0.05)(0.1 + 1.05256)
= 1.10769
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
=1.05256 + 0.05
2 [(0.05 + 1.05256) + (0.1 + 1.10769)]=
2nd approximation:
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(1)2 )]
∴ y2 = .... correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
=1.05256 + 0.05
2 [(0.05 + 1.05256) + (0.1 + 1.10769)]=
2nd approximation:
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(1)2 )]
∴ y2 = .... correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
=1.05256 + 0.05
2 [(0.05 + 1.05256) + (0.1 + 1.10769)]=
2nd approximation:
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(1)2 )]
∴ y2 = .... correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
Ex.: Using modified Euler’s method , find y(0.2)and y(0.4) given that
dy
dx= y + ex, y(0) = 0
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 anddy
dx= f(x, y) = y + ex
x1 = x0 + h = 0.2
y1 = y0 + hf(x0, y0)
= 0 + (0.2)(0 + e0)
= 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 anddy
dx= f(x, y) = y + ex
x1 = x0 + h = 0.2
y1 = y0 + hf(x0, y0)
= 0 + (0.2)(0 + e0)
= 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 anddy
dx= f(x, y) = y + ex
x1 = x0 + h = 0.2
y1 = y0 + hf(x0, y0)
= 0 + (0.2)(0 + e0)
= 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 anddy
dx= f(x, y) = y + ex
x1 = x0 + h = 0.2
y1 = y0 + hf(x0, y0)
= 0 + (0.2)(0 + e0)
= 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.2)] = 0.24214
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.2)] = 0.24214
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.2)] =
0.24214
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.2)] = 0.24214
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.2)] = 0.24214
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.2)] = 0.24214
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24214)] =
0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.2)] = 0.24214
2nd approximation:
y(2)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(1)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24635)] = 0.24678
4th approximation:
y(4)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(3)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24635)] = 0.24678
4th approximation:
y(4)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(3)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24635)] =
0.24678
4th approximation:
y(4)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(3)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24635)] = 0.24678
4th approximation:
y(4)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(3)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24635)] = 0.24678
4th approximation:
y(4)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(3)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24635)] = 0.24678
4th approximation:
y(4)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(3)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24678)] =
0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24635)] = 0.24678
4th approximation:
y(4)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(3)1 )]
= 0 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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Modified Euler’s Method
5th approximation:
y(5)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(4)1 )]
= 0.24682
∴ y1 = 0.24682 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 153: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/153.jpg)
Modified Euler’s Method
5th approximation:
y(5)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(4)1 )]
= 0.24682
∴ y1 = 0.24682 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 154: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/154.jpg)
Modified Euler’s Method
5th approximation:
y(5)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(4)1 )]
= 0.24682
∴ y1 = 0.24682 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 155: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/155.jpg)
Modified Euler’s Method
x2 = x1 + h = 0.4
y2 = y1 + hf(x1, y1)
= 0.24682 + (0.2)f(0.2, 0.24682)
= 0.54046
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 156: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/156.jpg)
Modified Euler’s Method
x2 = x1 + h = 0.4
y2 = y1 + hf(x1, y1)
= 0.24682 + (0.2)f(0.2, 0.24682)
= 0.54046
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 157: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/157.jpg)
Modified Euler’s Method
x2 = x1 + h = 0.4
y2 = y1 + hf(x1, y1)
= 0.24682 + (0.2)f(0.2, 0.24682)
= 0.54046
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 158: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/158.jpg)
Modified Euler’s Method
x2 = x1 + h = 0.4
y2 = y1 + hf(x1, y1)
= 0.24682 + (0.2)f(0.2, 0.24682)
= 0.54046
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 159: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/159.jpg)
Modified Euler’s Method
1st approximation:
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
= 0.59687
2nd approximation:
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(1)2 )]
=0.60251
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 160: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/160.jpg)
Modified Euler’s Method
1st approximation:
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
= 0.59687
2nd approximation:
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(1)2 )]
=0.60251
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 161: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/161.jpg)
Modified Euler’s Method
1st approximation:
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
= 0.59687
2nd approximation:
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(1)2 )]
=0.60251
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 162: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/162.jpg)
Modified Euler’s Method
1st approximation:
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
= 0.59687
2nd approximation:
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(1)2 )]
=0.60251
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 163: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/163.jpg)
Modified Euler’s Method
1st approximation:
y(1)2 = y1 +
h
2[f(x1, y1) + f(x2, y2)]
= 0.59687
2nd approximation:
y(2)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(1)2 )]
=0.60251
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 164: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/164.jpg)
Modified Euler’s Method
3rd approximation:
y(3)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(2)2 )]
= 0.60308
4th approximation:
y(4)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(3)2 )]
= 0.60313
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 165: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/165.jpg)
Modified Euler’s Method
3rd approximation:
y(3)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(2)2 )]
= 0.60308
4th approximation:
y(4)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(3)2 )]
= 0.60313
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 166: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/166.jpg)
Modified Euler’s Method
3rd approximation:
y(3)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(2)2 )]
= 0.60308
4th approximation:
y(4)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(3)2 )]
= 0.60313
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 167: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/167.jpg)
Modified Euler’s Method
3rd approximation:
y(3)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(2)2 )]
= 0.60308
4th approximation:
y(4)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(3)2 )]
= 0.60313
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 168: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/168.jpg)
Modified Euler’s Method
5th approximation:
y(5)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(4)2 )]
= 0.60314
∴ y2 = 0.60314 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 169: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/169.jpg)
Modified Euler’s Method
5th approximation:
y(5)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(4)2 )]
= 0.60314
∴ y2 = 0.60314 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
![Page 170: Numerical Methods - Oridnary Differential Equations - 2](https://reader031.vdocuments.mx/reader031/viewer/2022021919/587318501a28ab673e8b5ac3/html5/thumbnails/170.jpg)
Modified Euler’s Method
5th approximation:
y(5)2 = y1 +
h
2
[f(x1, y1) + f(x2, y
(4)2 )]
= 0.60314
∴ y2 = 0.60314 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2