![Page 1: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/1.jpg)
Numerical MethodsOrdinary Differential Equations - 3
Dr. N. B. Vyas
Department of Mathematics,Atmiya Institute of Technology & Science,
Rajkot (Gujarat) - [email protected]
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 2: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/2.jpg)
Ordinary Differential Equations
Runge’s Method: (Runge-Kutta Method of 3rd Order)Named after German mathematicians CARL RUNGE(1856-1927) and WILHELM KUTTA (1867-1944)
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2, y0 + k1
2
)k′ = hf (x0 + h, y0 + k1)
k3 = hf (x0 + h, y0 + k′)
Finally calculate
k = 16(k1 + 4k2 + k3)
required approximate value of y = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 3: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/3.jpg)
Ordinary Differential Equations
Runge’s Method: (Runge-Kutta Method of 3rd Order)Named after German mathematicians CARL RUNGE(1856-1927) and WILHELM KUTTA (1867-1944)
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2, y0 + k1
2
)k′ = hf (x0 + h, y0 + k1)
k3 = hf (x0 + h, y0 + k′)
Finally calculate
k = 16(k1 + 4k2 + k3)
required approximate value of y = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 4: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/4.jpg)
Ordinary Differential Equations
Runge’s Method: (Runge-Kutta Method of 3rd Order)Named after German mathematicians CARL RUNGE(1856-1927) and WILHELM KUTTA (1867-1944)
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2, y0 + k1
2
)
k′ = hf (x0 + h, y0 + k1)
k3 = hf (x0 + h, y0 + k′)
Finally calculate
k = 16(k1 + 4k2 + k3)
required approximate value of y = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 5: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/5.jpg)
Ordinary Differential Equations
Runge’s Method: (Runge-Kutta Method of 3rd Order)Named after German mathematicians CARL RUNGE(1856-1927) and WILHELM KUTTA (1867-1944)
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2, y0 + k1
2
)k′ = hf (x0 + h, y0 + k1)
k3 = hf (x0 + h, y0 + k′)
Finally calculate
k = 16(k1 + 4k2 + k3)
required approximate value of y = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 6: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/6.jpg)
Ordinary Differential Equations
Runge’s Method: (Runge-Kutta Method of 3rd Order)Named after German mathematicians CARL RUNGE(1856-1927) and WILHELM KUTTA (1867-1944)
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2, y0 + k1
2
)k′ = hf (x0 + h, y0 + k1)
k3 = hf (x0 + h, y0 + k′)
Finally calculate
k = 16(k1 + 4k2 + k3)
required approximate value of y = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 7: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/7.jpg)
Ordinary Differential Equations
Runge’s Method: (Runge-Kutta Method of 3rd Order)Named after German mathematicians CARL RUNGE(1856-1927) and WILHELM KUTTA (1867-1944)
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2, y0 + k1
2
)k′ = hf (x0 + h, y0 + k1)
k3 = hf (x0 + h, y0 + k′)
Finally calculate
k = 16(k1 + 4k2 + k3)
required approximate value of y = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 8: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/8.jpg)
Ordinary Differential Equations
Runge’s Method: (Runge-Kutta Method of 3rd Order)Named after German mathematicians CARL RUNGE(1856-1927) and WILHELM KUTTA (1867-1944)
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2, y0 + k1
2
)k′ = hf (x0 + h, y0 + k1)
k3 = hf (x0 + h, y0 + k′)
Finally calculate
k = 16(k1 + 4k2 + k3)
required approximate value of y = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 9: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/9.jpg)
Ordinary Differential Equations
Runge’s Method: (Runge-Kutta Method of 3rd Order)Named after German mathematicians CARL RUNGE(1856-1927) and WILHELM KUTTA (1867-1944)
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2, y0 + k1
2
)k′ = hf (x0 + h, y0 + k1)
k3 = hf (x0 + h, y0 + k′)
Finally calculate
k = 16(k1 + 4k2 + k3)
required approximate value of y = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 10: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/10.jpg)
Runge’s Method
Ex. Use Runge’s method to approximate y whenx = 1.1 given that y = 1.2 when x = 1 anddy
dx= 3x + y2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 11: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/11.jpg)
Runge’s Method
Sol. We havedy
dx= 3x + y2 , ∴ f(x, y) = 3x + y2
x0 = 1, y0 = 1.2 and h = 0.1
k1 = hf(x0, y0) = 0.444
k2 = hf(x0 + h
2 , y0 + k12
)= 0.51721
k′ = hf (x0 + h, y0 + k1) = 0.60027
k3 = hf (x0 + h, y0 + k′) = 0.65411
Hence k = 16(k1 + 4k2 + k3) = 0.5278
∴ approximate value of y = y0 + k = 1.7278
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 12: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/12.jpg)
Runge’s Method
Sol. We havedy
dx= 3x + y2 , ∴ f(x, y) = 3x + y2
x0 = 1, y0 = 1.2 and h = 0.1
k1 = hf(x0, y0) = 0.444
k2 = hf(x0 + h
2 , y0 + k12
)= 0.51721
k′ = hf (x0 + h, y0 + k1) = 0.60027
k3 = hf (x0 + h, y0 + k′) = 0.65411
Hence k = 16(k1 + 4k2 + k3) = 0.5278
∴ approximate value of y = y0 + k = 1.7278
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 13: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/13.jpg)
Runge’s Method
Sol. We havedy
dx= 3x + y2 , ∴ f(x, y) = 3x + y2
x0 = 1, y0 = 1.2 and h = 0.1
k1 = hf(x0, y0)
= 0.444
k2 = hf(x0 + h
2 , y0 + k12
)= 0.51721
k′ = hf (x0 + h, y0 + k1) = 0.60027
k3 = hf (x0 + h, y0 + k′) = 0.65411
Hence k = 16(k1 + 4k2 + k3) = 0.5278
∴ approximate value of y = y0 + k = 1.7278
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 14: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/14.jpg)
Runge’s Method
Sol. We havedy
dx= 3x + y2 , ∴ f(x, y) = 3x + y2
x0 = 1, y0 = 1.2 and h = 0.1
k1 = hf(x0, y0) = 0.444
k2 = hf(x0 + h
2 , y0 + k12
)
= 0.51721
k′ = hf (x0 + h, y0 + k1) = 0.60027
k3 = hf (x0 + h, y0 + k′) = 0.65411
Hence k = 16(k1 + 4k2 + k3) = 0.5278
∴ approximate value of y = y0 + k = 1.7278
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 15: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/15.jpg)
Runge’s Method
Sol. We havedy
dx= 3x + y2 , ∴ f(x, y) = 3x + y2
x0 = 1, y0 = 1.2 and h = 0.1
k1 = hf(x0, y0) = 0.444
k2 = hf(x0 + h
2 , y0 + k12
)= 0.51721
k′ = hf (x0 + h, y0 + k1)
= 0.60027
k3 = hf (x0 + h, y0 + k′) = 0.65411
Hence k = 16(k1 + 4k2 + k3) = 0.5278
∴ approximate value of y = y0 + k = 1.7278
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 16: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/16.jpg)
Runge’s Method
Sol. We havedy
dx= 3x + y2 , ∴ f(x, y) = 3x + y2
x0 = 1, y0 = 1.2 and h = 0.1
k1 = hf(x0, y0) = 0.444
k2 = hf(x0 + h
2 , y0 + k12
)= 0.51721
k′ = hf (x0 + h, y0 + k1) = 0.60027
k3 = hf (x0 + h, y0 + k′)
= 0.65411
Hence k = 16(k1 + 4k2 + k3) = 0.5278
∴ approximate value of y = y0 + k = 1.7278
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 17: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/17.jpg)
Runge’s Method
Sol. We havedy
dx= 3x + y2 , ∴ f(x, y) = 3x + y2
x0 = 1, y0 = 1.2 and h = 0.1
k1 = hf(x0, y0) = 0.444
k2 = hf(x0 + h
2 , y0 + k12
)= 0.51721
k′ = hf (x0 + h, y0 + k1) = 0.60027
k3 = hf (x0 + h, y0 + k′) = 0.65411
Hence k = 16(k1 + 4k2 + k3)
= 0.5278
∴ approximate value of y = y0 + k = 1.7278
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 18: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/18.jpg)
Runge’s Method
Sol. We havedy
dx= 3x + y2 , ∴ f(x, y) = 3x + y2
x0 = 1, y0 = 1.2 and h = 0.1
k1 = hf(x0, y0) = 0.444
k2 = hf(x0 + h
2 , y0 + k12
)= 0.51721
k′ = hf (x0 + h, y0 + k1) = 0.60027
k3 = hf (x0 + h, y0 + k′) = 0.65411
Hence k = 16(k1 + 4k2 + k3) = 0.5278
∴ approximate value of y = y0 + k
= 1.7278
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 19: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/19.jpg)
Runge’s Method
Sol. We havedy
dx= 3x + y2 , ∴ f(x, y) = 3x + y2
x0 = 1, y0 = 1.2 and h = 0.1
k1 = hf(x0, y0) = 0.444
k2 = hf(x0 + h
2 , y0 + k12
)= 0.51721
k′ = hf (x0 + h, y0 + k1) = 0.60027
k3 = hf (x0 + h, y0 + k′) = 0.65411
Hence k = 16(k1 + 4k2 + k3) = 0.5278
∴ approximate value of y = y0 + k = 1.7278
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 20: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/20.jpg)
Ordinary Differential Equations
Runge-Kutta Method of 2nd Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf (x0 + h, y0 + k1)
Find k =1
2(k1 + k2)
∴ y1 = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
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Ordinary Differential Equations
Runge-Kutta Method of 2nd Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf (x0 + h, y0 + k1)
Find k =1
2(k1 + k2)
∴ y1 = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 22: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/22.jpg)
Ordinary Differential Equations
Runge-Kutta Method of 2nd Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf (x0 + h, y0 + k1)
Find k =1
2(k1 + k2)
∴ y1 = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 23: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/23.jpg)
Ordinary Differential Equations
Runge-Kutta Method of 2nd Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf (x0 + h, y0 + k1)
Find k =1
2(k1 + k2)
∴ y1 = y0 + k
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 24: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/24.jpg)
Ordinary Differential Equations
Runge-Kutta Method of 2nd Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf (x0 + h, y0 + k1)
Find k =1
2(k1 + k2)
∴ y1 = y0 + kDr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 25: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/25.jpg)
Ordinary Differential Equations
Runge-Kutta Method of 4th Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2 , y0 + k12
)k3 = hf
(x0 + h
2 , y0 + k22
)k4 = hf(x0 + h, y0 + k3)
Find k =1
6(k1 + 2k2 + 2k3 + k4)
∴ y1 = y0 + k and x1 = x0 + h
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 26: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/26.jpg)
Ordinary Differential Equations
Runge-Kutta Method of 4th Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2 , y0 + k12
)k3 = hf
(x0 + h
2 , y0 + k22
)k4 = hf(x0 + h, y0 + k3)
Find k =1
6(k1 + 2k2 + 2k3 + k4)
∴ y1 = y0 + k and x1 = x0 + h
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 27: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/27.jpg)
Ordinary Differential Equations
Runge-Kutta Method of 4th Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2 , y0 + k12
)
k3 = hf(x0 + h
2 , y0 + k22
)k4 = hf(x0 + h, y0 + k3)
Find k =1
6(k1 + 2k2 + 2k3 + k4)
∴ y1 = y0 + k and x1 = x0 + h
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 28: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/28.jpg)
Ordinary Differential Equations
Runge-Kutta Method of 4th Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2 , y0 + k12
)k3 = hf
(x0 + h
2 , y0 + k22
)
k4 = hf(x0 + h, y0 + k3)
Find k =1
6(k1 + 2k2 + 2k3 + k4)
∴ y1 = y0 + k and x1 = x0 + h
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 29: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/29.jpg)
Ordinary Differential Equations
Runge-Kutta Method of 4th Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2 , y0 + k12
)k3 = hf
(x0 + h
2 , y0 + k22
)k4 = hf(x0 + h, y0 + k3)
Find k =1
6(k1 + 2k2 + 2k3 + k4)
∴ y1 = y0 + k and x1 = x0 + h
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 30: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/30.jpg)
Ordinary Differential Equations
Runge-Kutta Method of 4th Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2 , y0 + k12
)k3 = hf
(x0 + h
2 , y0 + k22
)k4 = hf(x0 + h, y0 + k3)
Find k =1
6(k1 + 2k2 + 2k3 + k4)
∴ y1 = y0 + k and x1 = x0 + h
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 31: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/31.jpg)
Ordinary Differential Equations
Runge-Kutta Method of 4th Order:
Consider the differential Equationdy
dx= f(x, y), y(x0) = y0
Calculate successively
k1 = hf(x0, y0)
k2 = hf(x0 + h
2 , y0 + k12
)k3 = hf
(x0 + h
2 , y0 + k22
)k4 = hf(x0 + h, y0 + k3)
Find k =1
6(k1 + 2k2 + 2k3 + k4)
∴ y1 = y0 + k and x1 = x0 + h
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
![Page 32: Numerical Methods - Oridnary Differential Equations - 3](https://reader033.vdocuments.mx/reader033/viewer/2022051318/587318501a28ab673e8b5ac1/html5/thumbnails/32.jpg)
Runge-Kutta Method of 2nd Order
Ex. Use Runge-kutta second order method to findthe approximate value of y(0.2) given thatdy
dx= x− y2 and y(0) = 1 and h = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
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Runge-Kutta Method of 2nd Order
Ex. Use 4th order Runge-kutta method to solvedy
dx= x2 + y2, y(0) = 1. Find y(0.2) with h = 0.1.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
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Runge-Kutta Method of 2nd Order
Ex. Determine y(0.1) and y(0.2) correct to four
decimal places fromdy
dx= 2x + y, y(0) = 1. Use
fourth order Runge-Kutta method
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3