 Solution

For $x>0$, use the Euler differential equation substitution $t=\ln x$, so $x=e^t$, $x{y}^{\prime }=\dot{y}$ and $x^2{y}^{\prime \prime }=\ddot{y}-\dot{y}$. The transformed equation isIts characteristic polynomial factors as $(r-1)(r+2)$. The forcing $e^t$ coincides with a homogeneous solution, so the method of undetermined coefficients requires a particular solution $y_p=At{e}^t$. Substitution gives $3A{e}^t=e^t$, hence $A=1/3$. Transforming back,The conditions at $x=1$ give $C_1+C_2=0$ and $C_1-2C_2+1/3=0$, soDifferentiating confirms both initial values and the original equation. The solution interval containing $x=1$ excludes the singular point $x=0$.