3.8.1.2 Example 2

Solve

\begin{align*} x^{2}y^{\prime \prime }+xy^{\prime }-9y & =0\\ y_{1} & =x^{3}\end{align*}

Putting the ode in normal form, it becomes

\[ y^{\prime \prime }+\frac {1}{x}y^{\prime }-\frac {9}{x^{2}}y=0 \]

Hence \(p=\frac {1}{x},q=-\frac {9}{x^{2}}\). Using EQ (1)

\begin{align*} v & =c_{1}\frac {e^{-\int pdx}}{y_{1}^{2}}\\ & =c_{1}\frac {e^{-\int \frac {1}{x}dx}}{x^{6}}\\ & =\frac {c_{1}}{x^{6}}e^{-\ln x}\\ & =c_{1}\frac {1}{x^{7}}\end{align*}

EQ (2) becomes

\begin{align*} u & =\int vdx\\ & =\int c_{1}x^{-7}dx\\ & =c_{1}\frac {x^{-6}}{-6}\\ & =c_{1}x^{-6}\end{align*}

(last step above just rewrites the constant). Hence the second solution is

\begin{align*} y_{2} & =y_{1}u\\ & =x^{3}\left ( c_{1}x^{-6}\right ) \\ & =c_{1}x^{-3}\end{align*}

Therefore the solution is

\begin{align*} y & =c_{3}y_{2}+c_{4}y_{1}\\ & =c_{1}\frac {1}{x^{3}}+c_{2}x^{3}\end{align*}

Where in last step above, constants were merged and renamed.