Example 3 \[ xy^{\prime \prime \prime }-y^{\prime \prime }=0 \] Let \(y^{\prime }=u\) then the ode becomes\[ xu^{\prime \prime }-u^{\prime }=0 \] Since \(u\) is missing then let \(u^{\prime }=v\) and the above becomes\[ xv^{\prime }-v=0 \] This is linear first order ode whose solution is \(v=c_{1}x\). Hence \(u^{\prime }=c_{1}x\). Integrating gives \(u=c_{1}x^{2}+c_{2}\). Hence \[ y^{\prime }=c_{1}x^{2}+c_{2}\] Integrating gives\[ y=c_{1}x^{3}+c_{2}x+c_{3}\]