3.5.6.2 Example 2

\[ \left ( y^{\prime }\right ) ^{\frac {3}{2}}=2y+3x+9 \]

Let \(u=2y+3x+9\) then \(u^{\prime }=2y^{\prime }+3\) then \(y^{\prime }=\frac {u^{\prime }-3}{2}\ \)and the ode becomes

\begin{align*} \left ( \frac {u^{\prime }-3}{2}\right ) ^{\frac {3}{2}} & =u\\ \left ( \left ( \frac {u^{\prime }-3}{2}\right ) ^{\frac {1}{2}}\right ) ^{3} & =u \end{align*}

Let \(\left ( \frac {u^{\prime }-3}{2}\right ) ^{\frac {1}{2}}=Y\) then

\begin{align*} Y^{3} & =u\\ Y & =\left \{ \begin {array} [c]{c}u^{\frac {1}{3}}\\ u^{\frac {1}{3}}\left ( -\frac {1}{2}+\frac {i\sqrt {3}}{2}\right ) ^{\frac {1}{3}}\\ u^{\frac {1}{3}}\left ( -\frac {1}{2}-\frac {i\sqrt {3}}{2}\right ) ^{\frac {1}{3}}\end {array} \right . \end{align*}

Hence

\begin{align*} \left ( \frac {u^{\prime }-3}{2}\right ) ^{\frac {1}{2}} & =\left \{ \begin {array} [c]{c}u^{\frac {1}{3}}\\ u^{\frac {1}{3}}\left ( -\frac {1}{2}+\frac {i\sqrt {3}}{2}\right ) ^{\frac {1}{3}}\\ u^{\frac {1}{3}}\left ( -\frac {1}{2}-\frac {i\sqrt {3}}{2}\right ) ^{\frac {1}{3}}\end {array} \right . \\ \left ( \frac {u^{\prime }-3}{2}\right ) & =\left \{ \begin {array} [c]{c}u^{\frac {2}{3}}\\ u^{\frac {2}{3}}\left ( -\frac {1}{2}+\frac {i\sqrt {3}}{2}\right ) ^{\frac {2}{3}}\\ u^{\frac {2}{3}}\left ( -\frac {1}{2}-\frac {i\sqrt {3}}{2}\right ) ^{\frac {2}{3}}\end {array} \right . \\ u^{\prime } & =\left \{ \begin {array} [c]{c}2u^{\frac {2}{3}}+3\\ 2u^{\frac {2}{3}}\left ( -\frac {1}{2}+\frac {i\sqrt {3}}{2}\right ) ^{\frac {2}{3}}+3\\ 2u^{\frac {2}{3}}\left ( -\frac {1}{2}-\frac {i\sqrt {3}}{2}\right ) ^{\frac {2}{3}}+3 \end {array} \right . \end{align*}

Each is solved as separable.

\[ \left \{ \begin {array} [c]{c}\int \frac {du}{2u^{\frac {2}{3}}+3}=\int dx\\ \int \frac {du}{2u^{\frac {2}{3}}\left ( -\frac {1}{2}+\frac {i\sqrt {3}}{2}\right ) ^{\frac {2}{3}}+3}=\int dx\\ \int \frac {du}{2u^{\frac {2}{3}}\left ( -\frac {1}{2}-\frac {i\sqrt {3}}{2}\right ) ^{\frac {2}{3}}+3}=\int dx \end {array} \right . \]

Hence the three solutions are

\[ \left \{ \begin {array} [c]{c}\int ^{2y\left ( x\right ) +3x+9}\frac {dz}{2z^{\frac {2}{3}}+3}=x+c_{1}\\ \int ^{2y\left ( x\right ) +3x+9}\frac {dz}{2z^{\frac {2}{3}}\left ( -\frac {1}{2}+\frac {i\sqrt {3}}{2}\right ) ^{\frac {2}{3}}+3}=x+c_{1}\\ \int ^{2y\left ( x\right ) +3x+9}\frac {dz}{2z^{\frac {2}{3}}\left ( -\frac {1}{2}-\frac {i\sqrt {3}}{2}\right ) ^{\frac {2}{3}}+3}=x+c_{1}\end {array} \right . \]