3.2.2 Example 2

$$\begin{array}{rl}{y}^{\prime }& =\frac{y}{x}\\ y\left(0\right)& =0\end{array}$$

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 $x\ne 0$. 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 $\mathrm{\infty }$ number of solutions.