Find the general solution for:

$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-4y=x$

From example $1,we$ know that considering the change $of$ variable:

$x=e^t,$ then $t=lnx,$ this differential equation becomes:

$\frac{d^{2}y}{d{t}^2}-4y=e^t$

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

solution for this differential equation:

$y_p(t)=-\frac{1}{3}{e}^t$

and the general solution for this differential equation $is$ given $by$:

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

therefore, $we$ conclude that the general solution for the original differential

equation $is$:

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