second order ode's (2b) · 6/14/2014 · second order odes (2b) 3 young won lim 6/15/14 first...
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Young Won Lim6/15/14
Second Order ODE's (2B)
Young Won Lim6/15/14
Copyright (c) 2011 - 2014 Young W. Lim.
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Second Order ODEs (2B) 3 Young Won Lim6/15/14
First Order ODE examples - solutions
d2 y
dx2+ 3
dydx
+ 2 y = 0d2 y
dx2+ 2
dydx
+ 1 y = 0d2 y
dx2+ 2
dydx
+ 2 y = 0
m2+ 3m + 2 = 0 m2
+ 2m + 1 = 0 m2+ 2m + 2 = 0
(m+2)(m+1) = 0 (m+1)2 = 0 (m+1−i)(m+1+i) = 0
m = −1, −2 m = −1 m = −1+ i , −1−i
e−1 x , e−2 x e−1 x , x e−1xe(−1+i) x , e(−1−i) x
y = c1e−1 x
+ c2e−2 x y = c1e
−1 x + c2 x e−1 xy = c1e
(−1+i) x+ c2e
(−1−i) x
y = e−x(c3 cos x + c 4sin x)
y = e−x (c1e+i x + c2e
−i x)
Second Order ODEs (2B) 4 Young Won Lim6/15/14
First Order ODE examples – verification (1)
d2 y
dx2+ 3
dydx
+ 2 y = 0d2 y
dx2+ 2
dydx
+ 1 y = 0d2 y
dx2+ 2
dydx
+ 2 y = 0
y = c1e−1 x
+ c2e−2 x y = c1e
−1 x+ c2 x e−1 x
y ' = −c1e−1 x − 2c2e
−2 x
3 y ' = −3c1e−1 x − 6c2e
−2 x
y ' ' = +c1e−1 x + 4c2e
−2 x
2 y = 2c1e−1 x + 2c2e
−2x y = c1e−1 x + c2 x e−1 x
y ' = −c1e−1 x
+ c2e−1x
− c2 x e−1 x
2 y ' = −2c1e−1x
+ 2c2e−1 x
− 2c2 x e−1 x
y ' ' = +c1e−1 x − c2e
−1 x
− c2e−1 x
+ c2 x e−1 x
y = c1e(−1+i) x
+ c2e(−1−i) x
2 y = 2c1e(−1+i)x
+ 2c2e(−1−i) x
y ' = (−1+i)c1e(−1+i)x
+ (−1−i)c2e(−1−i) x
2 y ' =(−2+2 i)c1 e(−1+ i) x
+ (−2−2 i)c2e(−1−i)x
y ' ' = (−1+i)2c1e(−1+i) x
+ (−1−i)2c2e(−1−i)x
=−2 i c1e(−1+i)x
+ 2 i c2e(−1−i) x
Second Order ODEs (2B) 5 Young Won Lim6/15/14
First Order ODE examples – verification (2)
d2 y
dx2+ 2
dydx
+ 2 y = 0
2 y = 2c1e(−1+i)x
+ 2c2e(−1−i) x
y ' = (−1+i)c1e(−1+i)x + (−1−i)c2e
(−1−i) x
2 y ' =(−2+2 i)c1 e(−1+ i) x + (−2−2 i)c2e
(−1−i)x
y ' ' = (−1+i)2c1e(−1+i) x + (−1−i)2c2e
(−1−i)x
=−2 i c1e(−1+i)x
+ 2 i c2e(−1−i) x
y = e−x (c3cos x + c4 sin x)
y ' ' + 2 y ' + 2 y = 0
y = e−x(c3 cos x + c 4sin x)
y = e−x (c1e+i x + c2e
−i x)
Second Order ODEs (2B) 6 Young Won Lim6/15/14
First Order ODE examples – verification (3)
d2 y
dx2+ 2
dydx
+ 2 y = 0
2 y = e−x(2c3cos x + 2c4 sin x)
y ' ' =−e−x((−c3 + c4)cos x −(c3 + c4)sin x) + e−x
((c3 − c4)sin x −(c3 + c4)cos x)
y ' =−e−x(c3cos x + c4sin x) + e−x(−c3 sin x + c4cos x)
y = e−x (c3cos x + c4 sin x)
2 y ' = e−x(2(−c3 + c4)cos x − 2(c3 + c4)sin x)
= e−x((−c3 + c4)cos x − (c3 + c4)sin x)
= e−x((c3 − c 4 − c3 − c4)cos x + (c3 + c4 + c3 − c4)sin x )
y ' ' = e−x(−2c4cos x + 2c3 sin x ) y ' ' + 2 y ' + 2 y = 0
y = e−x(c3 cos x + c 4sin x)
y = e−x (c1e+i x + c2e
−i x)
Second Order ODEs (2B) 7 Young Won Lim6/15/14
Fundamental Set Examples
C1
Second Order EQ
ad2 yd x2 + b
d yd x
+ c y = 0
y1 y2+ C2
y1
y2
C1e(α+iβ) x + C2 e
(α−iβ) x
c3eα x cos(β x) + c4 e
α x sin(β x)
= eα x(c3 cos(β x) + c4 sin(β x))
e(α+iβ) x
e(α−iβ) x
{e(α+iβ) x+ e(α+iβ) x
}/2
{e(α+iβ) x+ i e(α+ iβ) x
}/2 i
= eα xcos(β x)
= eα xsin (β x)
12
y1 y2+ 12
y3 =
12 i
y1 y2− 12 i
y4 =
c3 y3 y4+ c3
Second Order ODEs (2B) 8 Young Won Lim6/15/14
Complex Exponentials
e+iβ t
e−iβ t
e0
eπ/4
eπ/ 2
e3π/ 4
eπ
e5π/4
e3π/2
e7π/4
e0
e−7π /4
e−3π /2
e−5π /4
e−π
e−3π /4
e−π /2
e−π /4
Second Order ODEs (2B) 9 Young Won Lim6/15/14
First Order ODE examples (1)
d2 y
dx2+ 3
dydx
+ 2 y = 0d2 y
dx2+ 2
dydx
+ 1 y = 0d2 y
dx2+ 2
dydx
+ 2 y = 0
y = c1e−1 x
+ c2e−2 x y = c1e
−1 x+ c2 x e−1 x
y = e−x(c3 cos x + c 4sin x)
2 y = 2(e−1 x+ e−2 x
)
3 y ' = 3(−e−1 x− 2e−2 x
)
y ' ' = +e−1 x + 4 e−2x
y = (e−1 x+ e−2 x
)
Second Order ODEs (2B) 10 Young Won Lim6/15/14
First Order ODE examples (1)
d2 y
dx2+ 3
dydx
+ 2 y = 0d2 y
dx2+ 2
dydx
+ 1 y = 0d2 y
dx2+ 2
dydx
+ 2 y = 0
y = c1e−1 x
+ c2e−2 x y = c1e
−1 x+ c2 x e−1 x
y = e−x(c3 cos x + c 4sin x)
2 y ' = 2(−x e−x)
y ' ' = −e−x + x e−x
y = (e−x+ x e−x
)
Second Order ODEs (2B) 11 Young Won Lim6/15/14
First Order ODE examples (1)
d2 y
dx2+ 3
dydx
+ 2 y = 0d2 y
dx2+ 2
dydx
+ 1 y = 0d2 y
dx2+ 2
dydx
+ 2 y = 0
y = c1e−1 x
+ c2e−2 x y = c1e
−1 x+ c2 x e−1 x
y = e−x(c3 cos x + c 4sin x)
2 y ' = 2e−x(cos x + sin x)
y ' ' = 2(−e−x cos x − e−xsin x)
y = e−x(cos x + sin x )
+ 2e−x (−sin x + cos x)
= 2⋅2e−x cos x
2 y = 2e−x(cos x + sin x)
Second Order ODEs (2B) 12 Young Won Lim6/15/14
cos(x) + sin(x)
Second Order ODEs (2B) 13 Young Won Lim6/15/14
e-x (cos(x) + sin(x))
Second Order ODEs (2B) 14 Young Won Lim6/15/14
e-x (cos(x) + sin(x))
Second Order ODEs (2B) 15 Young Won Lim6/15/14
e-x (cos(x) + sin(x))
Second Order ODEs (2B) 16 Young Won Lim6/15/14
x e-x
Second Order ODEs (2B) 17 Young Won Lim6/15/14
x e-x
Second Order ODEs (2B) 18 Young Won Lim6/15/14
x e-x
Second Order ODEs (2B) 19 Young Won Lim6/15/14
Diff & Integrate xex
Second Order ODEs (2B) 20 Young Won Lim6/15/14
Diff & Integrate x2ex
Second Order ODEs (2B) 21 Young Won Lim6/15/14
Diff & Integrate x3ex
Second Order ODEs (2B) 22 Young Won Lim6/15/14
Differentiate x10ex
Second Order ODEs (2B) 23 Young Won Lim6/15/14
Integrate x10ex
Second Order ODEs (2B) 24 Young Won Lim6/15/14
ODE y'' -2y' +y = 0
Second Order ODEs (2B) 25 Young Won Lim6/15/14
ODE y'' -2y' +y = ex
Second Order ODEs (2B) 26 Young Won Lim6/15/14
xiex => y'' -2y' +y
Second Order ODEs (2B) 27 Young Won Lim6/15/14
ODE y'' -2y' +y = xiex
Second Order ODEs (2B) 28 Young Won Lim6/15/14
ODE y'' -2y' +y = cosi(x)ex
Second Order ODEs (2B) 29 Young Won Lim6/15/14
ODE y'' -2y' +y = eix, xi
Second Order ODEs (2B) 30 Young Won Lim6/15/14
Young Won Lim6/15/14
References
[1] http://en.wikipedia.org/[2] M.L. Boas, “Mathematical Methods in the Physical Sciences”[3] E. Kreyszig, “Advanced Engineering Mathematics”[4] D. G. Zill, W. S. Wright, “Advanced Engineering Mathematics”