Page 135 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Solution: The given equation is

(D 2 3D 2)y = e 5x
Auxiliary equation is

m 2 3m 2 = 0 or (m 1) (m 2) =0
m = 1, 2

C.F. = c e x c e 2x
1 2
1 5x
P.I. = 2 e
D 3D 2
1 5x 1 5x
= e e
25 3.5 2 12
y = C.F. + P.I.

1 5x
x
2x
= c e c e e
2
1
12
2
d y dy
Example 2: Solve: y e x .
dx 2 dx
Solution: Here the auxiliary equation is
1 3
2
m + m + 1 = 0, m i
2 2
1
x 1 1
C.F. = e 2 A cos 3x B sin 3x
2 2
1 x
Also P.I. = 2 e
D D 1
1 x x
= 2 e e
( 1) ( 1) 1

Hence the general solution of the given equation is
1
x 3 3
y = e 2 A cos x B sin x e x
2 2

Self Assessment

Solve the following differential equations:

7. (D 2 5D 6)y e 2x .
8. (D 3 D 2 4D 4)y e 3 x .

9. (4D 2 4D 3)y e 2 x

10. (D 3 1)y (e x 1) 2

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