\begin {align*} y^{\prime } & =\frac {y}{x}\\ y\left ( 0\right ) & =1 \end {align*}
In standard form \(y^{\prime }-p\left ( x\right ) y=q\left ( x\right ) \). So \(p=\frac {-1}{x},q=0\). Hence the domain of \(p\) is all \(x\) except \(x=0\). Domain of \(q\) is all \(x\). Since the IC includes \(x=0\) then no guarantee solution exists or be unique. Theory does not say anything. We have to try to solve the ode to find out. Solving gives\[ y=cx \] As solution. Applying I.C. gives\[ 1=0 \] Not possible. Therefore no solution exist.