 Solution

Set $v=y^{\prime }$ to reduce the second-order linear differential equation to $x{v}^{\prime }+v=0$. Equivalently, $(xv{)}^{\prime }=0$, so $v=C_1/x$ on any interval excluding zero. The derivative condition at $x=1$ gives $C_1=2$.

Integrating on the positive interval containing the initial point yields $y=2\ln x+C_2$. The value $y(1)=-1$ then fixes $C_2=-1$, givingIts derivative is $2/x$, and direct substitution verifies the equation. The selected solution diverges as $x\downarrow 0$ and cannot extend through zero as a finite-valued function.