5.2.2.2 Example 2

\[ xy^{\prime \prime \prime \prime }+y^{\prime \prime \prime }+y^{\prime \prime }=0 \]

Let \(y^{\prime }=u\) then the ode becomes

\[ xu^{\prime \prime \prime }+u^{\prime \prime }+u^{\prime }=0 \]

Since \(u\) is missing then let \(u^{\prime }=v\) and the above becomes

\[ xv^{\prime \prime }+v^{\prime }+v=0 \]

This is now second order ode. This is Bessel ode whose solution is

\[ v=c_{3}\operatorname {BesselJ}_{0}\left ( 2\sqrt {x}\right ) +c_{4}\operatorname {BesselY}_{0}\left ( 2\sqrt {x}\right ) \]

Hence

\[ u^{\prime }=c_{3}\operatorname {BesselJ}_{0}\left ( 2\sqrt {x}\right ) +c_{4}\operatorname {BesselY}_{0}\left ( 2\sqrt {x}\right ) \]

This is solved by quadrature giving

\[ u=c_{3}\sqrt {x}\operatorname {BesselJ}_{1}\left ( 2\sqrt {x}\right ) +c_{4}\sqrt {x}\operatorname {BesselY}_{1}\left ( 2\sqrt {x}\right ) +c_{2}\]

Hence

\[ y^{\prime }=c_{3}\sqrt {x}\operatorname {BesselJ}_{1}\left ( 2\sqrt {x}\right ) +c_{4}\sqrt {x}\operatorname {BesselY}_{1}\left ( 2\sqrt {x}\right ) +c_{2}\]

This is solved by quadrature giving

\[ y=c_{3}x\operatorname {BesselJ}_{2}\left ( 2\sqrt {x}\right ) +c_{4}x\operatorname {BesselY}_{2}\left ( 2\sqrt {x}\right ) +c_{2}x+c_{1}\]