Question

Find the general solution of the nonhomogeneous differential equation x^2y''-2y=3(x^2) -1, (x>0).

(I have already verified that y1=1/x is a solution of the differential equation x^2y''-2y=0).

Answer 1

G.S=
`C_1(1)/(x)+C_2x^2+x^2logx+(1)/(2)`

Explanation:

We are given that non-homogeneous differential equation

`x^2y''-2y=3(x^2)-1`

It is Cauchy Euler equation

Substitute x=e^t x>0

Auxillary equation

`D'(D'-1)-2=0`

`D'^2-D'-2=0`

`(D'-2)(D'+1)=0`

`D'-2=0 \implies D'=2`

`D'+1=0\implies D'=-1`

Complementary solution

`y=C_1e^(-t)+C_2e^(2t)`

`y=C_1(1)/(x)+C_2x^2`

Particular solution

`y_p=(3e^(2t))/(D'^2-D'-2)-(e^(0t))/(D'^2-D'-2)`

`y_p=te^(2t)+(1)/(2)=x^2logx+(1)/(2)`

G.S=
`C_1(1)/(x)+C_2x^2+x^2logx+(1)/(2)`

Hence, general solution G.S=
`C_1(1)/(x)+C_2x^2+x^2logx+(1)/(2)`