Question

(1) solve the following differential equation
$({d}^{6} -{d}^{4})y ={x}^{2}$
​

Answer 1

Answer:

Answer

We have to solve (D

2

−4D+1)y=x

2

The characteristic equation is p

2

−4p+1=0

⇒p=

2

4±

16−4

=

2

4±2

3

=2±

3

Thus Complementary function C.F.=Ae

(2+

3

)x

+Be

(2−

3

)x

Particular integral P.I.=

D

2

−4D+1

1

(x

2

)

=[1−(4D−D

2

)]

−1

(x

2

)

=[1+(4D−D

2

)+(4D−D

2

)

2

+...](x

2

)

=[1+4D+15D

2

+...](x

2

)

∴P.I.=x

2

+8x+30

Hence the general solution is y=C.F.+P.I.

y.=Ae

(2+

3

)x

+Be

(2−

3

)x

+(x

2

+8x+30)