Find the general solution for:

$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-y=x^2-7$

Consider the change $of$ variable:

$x=e^t,$ then $t=lnx,$ then

the original equation becomes:

$\frac{d^{2}y}{d{t}^2}-y=e^{2t}-7$

and the general solution for the homogeneous differential equation

$\frac{d^{2}y}{d{t}^2}-y=0$

$is$ given $by$:

$c_1{e}^{-t}+c_2{e}^t$

$by$ using the method $of$ undetermined coefficients, $we$ obtain a particular

solution for the differential equation:

$\frac{d^{2}y}{d{t}^2}-y=e^{2t}-7$

$as{y}_p(t)=A{e}^{2t}+B.$

Replacing $in$ the differential equation $we$ obtain: $A=\frac{1}{3}$ and $B=7.$

therefore, the general solution $is$:

$y(t)=c_1{e}^{-t}+c_2{e}^t+\frac{1}{3}{e}^{2t}+7$

and the general solution for the original differential equation $is$:

$y(x)=c_1{x}^{-1}+c_{2}x+\frac{1}{3}{x}^2+7$