Example 1 \[ x^{2}y^{\prime \prime \prime }+xy^{\prime \prime }+y^{\prime }=0 \] This is not Euler type as it stands. Let \(y^{\prime }=u\) then the ode becomes\[ x^{2}u^{\prime \prime }+xu^{\prime }+u=0 \] This is now Euler type. Solving it gives\[ u=c_{2}\cos \left ( \ln x\right ) +c_{3}\sin \left ( \ln x\right ) \] Hence\[ y^{\prime }=c_{2}\cos \left ( \ln x\right ) +c_{3}\sin \left ( \ln x\right ) \] Solving this as first order ode of quadrature type gives\begin {align*} y & =\frac {c_{2}}{2}x\cos \left ( \ln x\right ) +\frac {c_{2}}{2}x\sin \left ( \ln x\right ) -\frac {1}{2}c_{3}x\cos \left ( \ln x\right ) +\frac {1}{2}c_{3}x\sin \left ( \ln x\right ) +c_{1}\\ & =x\cos \left ( \ln x\right ) \left ( \frac {c_{2}}{2}-\frac {1}{2}c_{3}\right ) +x\sin \left ( \ln x\right ) \left ( \frac {c_{2}}{2}++\frac {1}{2}c_{3}\right ) +c_{1}\\ & =C_{2}x\cos \left ( \ln x\right ) +C_{3}x\sin \left ( \ln x\right ) +c \end {align*}