Example 1

\begin{align*} y^{\prime }-\frac {1}{2\sqrt {x}}y & =x\\ y\left ( 0\right ) & =1 \end{align*}

In normal form the ode is

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

Hence here we have \(p\left ( x\right ) =\frac {-1}{2\sqrt {x}}\) and \(q\left ( x\right ) =x\). The domain of \(p\left ( x\right ) \) is all the real line except \(x=0\) and domain of \(q\left ( x\right ) \) is all the real line. Combining domains gives all the real line except \(x=0\). Since initial \(x_{0}\) is \(x=0\) which is outside the domain, then uniqueness and existence theory do not apply. Solving gives

\[ y=-2x^{\frac {3}{2}}-12\sqrt {x}-6x-12+c_{1}e^{\sqrt {x}}\]

Applying IC

\begin{align*} 1 & =-12+c_{1}\\ c_{1} & =13 \end{align*}

Hence solution is

\[ y=-2x^{\frac {3}{2}}-12\sqrt {x}-6x-12+13e^{\sqrt {x}}\qquad x\neq 0 \]

In this case, solution exists and unique.