3.2.1 Example 1

$$\begin{array}{rl}{y}^{\prime }& =\frac{y}{x}\\ y\left(0\right)& =1\end{array}$$

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.