2.1.1.1 Example

\begin{align*} xy^{\prime \prime }+y^{\prime }+3y & =\sin \left ( x\right ) \\ y\left ( 0\right ) & =0\\ y^{\prime }\left ( 0\right ) & =1 \end{align*}

In standard form

\[ y^{\prime \prime }+\frac {1}{x}y^{\prime }+\frac {3}{x}y=\frac {1}{x}\sin x \]

We see that \(p\left ( x\right ) =\frac {1}{x}\) is not continuous at \(x_{0}=0\). Hence theorem does not apply. It turns out that there is no solution to this ode with these initial conditions. Changing \(x_{0}\) to \(1\) instead of zero, solution exists and is unique.