\begin {align*} y^{\prime }-\frac {y}{x} & =0\\ y\left ( 0\right ) & =1 \end {align*}
In normal form the ode is \[ y^{\prime }+p\left ( x\right ) y=q\left ( x\right ) \] The above shows that \(p\left ( x\right ) =-\frac {1}{x}\).The domain of \(p\left ( x\right ) \) is all the real line except \(x=0\). Since initial \(x_{0}\) is \(x=0\) then uniqueness and existence theory do not apply. We are not guaranteed solution exist or if it exist, is unique. Solving gives\[ y=c_{1}x \] Applying IC gives\[ 1=0 \] Which is not possible. Hence no solution exist.