Page 131 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 2x
or y = e c 1 cosh 3x c 2 sin h 3x .

3
2
d y d y dy
Example 3: Solve: 3 2 2 4 8y 0.
dx dx dx
Solution: The given equation is
3
(D 2D 2 4D + 8) y = 0
Auxiliary equation is
2
2
m 2m 4m + 8 = 0
2
or (m 2) (m 4) = 0; m = 2, 2.
General solution is

y = (c 1 c 2 ) x e 2x c e 2x .
3
2
d y
Example 4: Solve: 4y 0.
dx 2
Solution: The given equation is

2
(D + 4) y = 0.
Auxiliary equation is

2
m + 4 = 0 or m = 2i.
The general solution is

y = c cos 2x + c sin 2x.
1 2
Self Assessment

3. Solve

3
2
d y d y dy
9 23 15y 0
dx 3 dx 2 dx
4. Solve

2
d y 8 dy 25y 0
dx 2 dx

5. Solve
2
3
d y d y dy
4 5 2y 0
dx 3 dx 2 dx
6. Solve

4
2
3
d y 2 d y 5 d y 8 dy 4y 0
dx 4 dx 3 dx 2 dx

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