3.5.5.1 Example 1

\begin{align*} \left ( y^{\prime }\right ) ^{4}+f\left ( x\right ) \left ( y-a\right ) ^{3}\left ( y-b\right ) ^{3}\left ( y-c\right ) ^{2} & =0\\ \left ( y^{\prime }\right ) ^{4} & =-f\left ( x\right ) \left ( y-a\right ) ^{3}\left ( y-b\right ) ^{3}\left ( y-c\right ) ^{2}\\ \frac {\left ( y^{\prime }\right ) ^{4}}{\left ( y-a\right ) ^{3}\left ( y-b\right ) ^{3}\left ( y-c\right ) ^{2}} & =-f\left ( x\right ) \\ \left ( \frac {y^{\prime }}{\left ( \left ( y-a\right ) ^{3}\left ( y-b\right ) ^{3}\left ( y-c\right ) ^{2}\right ) ^{\frac {1}{4}}}\right ) ^{4} & =-f\left ( x\right ) \\ \frac {y^{\prime }}{\left ( \left ( y-a\right ) ^{3}\left ( y-b\right ) ^{3}\left ( y-c\right ) ^{2}\right ) ^{\frac {1}{4}}} & =\left ( -f\left ( x\right ) \right ) ^{\frac {1}{4}}\\ \frac {y^{\prime }}{\left ( \left ( y-a\right ) \left ( y-b\right ) \left ( y-c\right ) ^{\frac {2}{3}}\right ) ^{\frac {3}{4}}} & =\left ( -f\left ( x\right ) \right ) ^{\frac {1}{4}}\\ \frac {dy}{\left ( \left ( y-a\right ) \left ( y-b\right ) \left ( y-c\right ) ^{\frac {2}{3}}\right ) ^{\frac {3}{4}}} & =\left ( -f\left ( x\right ) \right ) ^{\frac {1}{4}}dx\\ \int ^{y\left ( x\right ) }\frac {1}{\left ( \left ( z-a\right ) \left ( z-b\right ) \left ( z-c\right ) ^{\frac {2}{3}}\right ) ^{\frac {3}{4}}}dz & =\int ^{x}\left ( -f\left ( \tau \right ) \right ) ^{\frac {1}{4}}d\tau +c_{1}\end{align*}