Find the general solution for:

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

Consider the change $of$ variable:

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

the original equation becomes:

$\frac{d^{2}y}{d{t}^2}+\frac{dy}{dt}-2y=e^t$

and the general solution for the homogeneous differential equation

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

$is$ given $by$:

$c_1{e}^{-2t}+c_2{e}^t$

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

solution for the differential equation:

$\frac{d^{2}y}{d{t}^2}+\frac{dy}{dt}-2y=c^t$

$as{y}_p(t)=\frac{1}{3}t{e}^t$

therefore, the general solution $is$:

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

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

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