3.5.5.4 Example 4

\[ \left ( y^{\prime }\right ) ^{\frac {1}{3}}=yx \]

For this form, we write \(y^{\prime }=\left ( yx\right ) ^{3}\) but this is always with the assumption that \(yx>0\).

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