5.2.2.3 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

\[ y=c_{1}x^{3}+c_{2}x+c_{3}\]

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