2.5.1.2 Example 2 \(y^{\prime }-\frac {y}{x}=0\)

\begin{align*} y^{\prime }-\frac {y}{x} & =0\\ y\left ( 0\right ) & =1 \end{align*}

In normal form the ode is

\[ y^{\prime }+p\left ( x\right ) y=q\left ( x\right ) \]

The above shows that \(p\left ( x\right ) =-\frac {1}{x}\).The domain of \(p\left ( x\right ) \) is all the real line except \(x=0\). Since initial \(x_{0}\) is \(x=0\) then uniqueness and existence theory do not apply. We are not guaranteed solution exist or if it exist, is unique. Solving gives

\[ y=c_{1}x \]

Applying IC gives

\[ 1=0 \]

Which is not possible. Hence no solution exist.