3.5.5.2 Example 2

\begin{align*} \left ( y^{\prime }\right ) ^{3} & =y\sin x\\ \frac {\left ( y^{\prime }\right ) ^{3}}{y} & =\sin x\\ \left ( \frac {y^{\prime }}{y^{\frac {1}{3}}}\right ) ^{3} & =\sin x \end{align*}

Hence we have 3 solutions

\begin{align*} \frac {y^{\prime }}{y^{\frac {1}{3}}} & =\left \{ \begin {array} [c]{c}\sin ^{\frac {1}{3}}x\\ -\left ( -1\right ) ^{\frac {1}{3}}\sin ^{\frac {1}{3}}x\\ \left ( -1\right ) ^{\frac {2}{3}}\sin ^{\frac {1}{3}}x \end {array} \right . \\ \frac {dy}{y^{\frac {1}{3}}} & =\left \{ \begin {array} [c]{c}\sin ^{\frac {1}{3}}xdx\\ -\left ( -1\right ) ^{\frac {1}{3}}\sin ^{\frac {1}{3}}xdx\\ \left ( -1\right ) ^{\frac {2}{3}}\sin ^{\frac {1}{3}}xdx \end {array} \right . \\ \int \frac {dy}{y^{\frac {1}{3}}} & =\left \{ \begin {array} [c]{c}\int \sin ^{\frac {1}{3}}xdx\\ -\left ( -1\right ) ^{\frac {1}{3}}\int \sin ^{\frac {1}{3}}xdx\\ \left ( -1\right ) ^{\frac {2}{3}}\int \sin ^{\frac {1}{3}}xdx \end {array} \right . \\ \frac {3}{2}y^{\frac {2}{3}} & =\left \{ \begin {array} [c]{c}\int \sin ^{\frac {1}{3}}xdx+c_{1}\\ -\left ( -1\right ) ^{\frac {1}{3}}\int \sin ^{\frac {1}{3}}xdx+c_{1}\\ \left ( -1\right ) ^{\frac {2}{3}}\int \sin ^{\frac {1}{3}}xdx+c_{1}\end {array} \right . \\ y^{\frac {2}{3}} & =\left \{ \begin {array} [c]{c}\frac {2}{3}\int \sin ^{\frac {1}{3}}xdx+c_{1}\\ -\frac {2}{3}\left ( -1\right ) ^{\frac {1}{3}}\int \sin ^{\frac {1}{3}}xdx+c_{1}\\ \frac {2}{3}\left ( -1\right ) ^{\frac {2}{3}}\int \sin ^{\frac {1}{3}}xdx+c_{1}\end {array} \right . \\ y & =\left \{ \begin {array} [c]{c}\left ( \frac {2}{3}\int \sin ^{\frac {1}{3}}xdx+c_{1}\right ) ^{\frac {3}{2}}\\ \left ( -\frac {2}{3}\left ( -1\right ) ^{\frac {1}{3}}\int \sin ^{\frac {1}{3}}xdx+c_{1}\right ) ^{\frac {3}{2}}\\ \left ( \frac {2}{3}\left ( -1\right ) ^{\frac {2}{3}}\int \sin ^{\frac {1}{3}}xdx+c_{1}\right ) ^{\frac {3}{2}}\end {array} \right . \end{align*}