\begin {align*} y^{\prime \prime }+\frac {1}{x-1}y^{\prime }+3y & =x\\ y\left ( 1\right ) & =0\\ y^{\prime }\left ( 1\right ) & =1 \end {align*}
In standard form\[ y^{\prime \prime }+py^{\prime }+qy=f \] \(p\left ( x\right ) =\frac {1}{x-1}\) is not continuous at \(x_{0}=1\). Hence theorem does not apply. It turns out that there is no solution to this ode with these initial conditions. Changing \(x_{0}\) to \(0\) instead then a solution exists and is unique.