Question

The Solution of differential equation D²y - 9x = 0​

Answer 1

Explanation:

This is second order non homogeneous differential equation.

Its solution is given as CF+PI

For CF part

dx

2

d

2

y

+3y=0....(ii)

y=c

1

e

mx

dx

dy

=c

1

me

mx

dx

2

d

2

y

=c

1

m

2

e

mx

Substituting in (ii), we get

m

2

e

mx

+3e

mx

=0

e

mx

(m

2

+3)=0

m

2

+3=0

⇒m=

−3

m=i

3

y=c

1

e

i

3

x

=c

1

(cos

3

x+isin

3

x)

y=c

1

cos

3

x+c

2

sin

3

x

For PI part

Let y=cx+d

dx

dy

=c

dx

2

d

2

y

=0

Substituting in (i)

0+3(cx+d)=−2x

3cx+3d=−2x

Comparing both sides

c=−

3

2

,d=0

So solution for PI part is

y=−

3

2

x

General solution is CF+PI

c

1

cos

3

x+c

2

sin

3

x−

3

2

x