Example 2

\begin{align*} y^{\prime } & =\frac {y}{x}\\ y\left ( 0\right ) & =0 \end{align*}

In standard form \(y^{\prime }-p\left ( x\right ) y=q\left ( x\right ) \). So \(p=\frac {-1}{x},q=0\). Domain of \(p\) is \(x\neq 0\). Domain of \(q\) is all \(x\). Since IC includes \(x=0\) then theory says nothing about existence and uniqueness. We have to solve the ode to find out. Solving gives

\[ y=cx \]

Applying I.C. gives

\[ 0=0 \]

Which is true for any \(c\). Hence solution exist which is \(y=cx\) for any \(c\). Hence solution is not unique. There are \(\infty \) number of solutions.