Example 1

\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.