3.5.5.3 Example 3

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

Hence we have 3 solutions

\begin{align*} \frac {y^{\prime }}{y^{\frac {1}{3}}} & =\left \{ \begin {array} [c]{c}x^{\frac {1}{3}}\\ -\left ( -1\right ) ^{\frac {1}{3}}x^{\frac {1}{3}}\\ \left ( -1\right ) ^{\frac {2}{3}}x^{\frac {1}{3}}\end {array} \right . \\ \frac {dy}{y^{\frac {1}{3}}} & =\left \{ \begin {array} [c]{c}x^{\frac {1}{3}}dx\\ -\left ( -1\right ) ^{\frac {1}{3}}x^{\frac {1}{3}}xdx\\ \left ( -1\right ) ^{\frac {2}{3}}x^{\frac {1}{3}}xdx \end {array} \right . \\ \int \frac {dy}{y^{\frac {1}{3}}} & =\left \{ \begin {array} [c]{c}\int x^{\frac {1}{3}}dx\\ -\left ( -1\right ) ^{\frac {1}{3}}\int x^{\frac {1}{3}}dx\\ \left ( -1\right ) ^{\frac {2}{3}}\int x^{\frac {1}{3}}dx \end {array} \right . \\ \frac {3}{2}y^{\frac {2}{3}} & =\left \{ \begin {array} [c]{c}\frac {3}{4}x^{\frac {4}{3}}+c_{1}\\ -\left ( -1\right ) ^{\frac {1}{3}}\left ( \frac {3}{4}x^{\frac {4}{3}}\right ) +c_{1}\\ \left ( -1\right ) ^{\frac {2}{3}}\left ( \frac {3}{4}x^{\frac {4}{3}}\right ) +c_{1}\end {array} \right . \\ y^{\frac {2}{3}} & =\left \{ \begin {array} [c]{c}\frac {1}{2}x^{\frac {4}{3}}+c_{1}\\ -\left ( -1\right ) ^{\frac {1}{3}}\left ( \frac {1}{2}x^{\frac {4}{3}}\right ) +c_{1}\\ \left ( -1\right ) ^{\frac {2}{3}}\left ( \frac {1}{2}x^{\frac {4}{3}}\right ) +c_{1}\end {array} \right . \\ y & =\left \{ \begin {array} [c]{c}\left ( \frac {1}{2}x^{\frac {4}{3}}+c_{1}\right ) ^{\frac {3}{2}}\\ \left ( -\left ( -1\right ) ^{\frac {1}{3}}\left ( \frac {1}{2}x^{\frac {4}{3}}\right ) +c_{1}\right ) ^{\frac {3}{2}}\\ \left ( \left ( -1\right ) ^{\frac {2}{3}}\left ( \frac {1}{2}x^{\frac {4}{3}}\right ) +c_{1}\right ) ^{\frac {3}{2}}\end {array} \right . \end{align*}