= Solution
For $x>0$, use the <Euler differential equation> substitution $t=\ln x$, so $x=e^t$, $xy'=\dot y$ and $x^2y''=\ddot y-\dot y$. The transformed equation is
$$
\ddot y+\dot y-2y=e^t.
$$
Its <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 $3Ae^t=e^t$, hence $A=1/3$. Transforming back,
$$
y(x)=C_1x+C_2x^{-2}+\frac{x\ln x}{3}.
$$
The conditions at $x=1$ give $C_1+C_2=0$ and $C_1-2C_2+1/3=0$, so
$$
\boxed{y(x)=\frac{x\ln x}{3}-\frac{x}{9}+\frac1{9x^2},\qquad x>0.}
$$
Differentiating confirms both initial values and the original equation. The solution interval containing $x=1$ excludes the singular point $x=0$.
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