3.3.4.10 Example 10

\[ y^{\prime }=ay^{\frac {a-1}{a}}\]

Integrating gives

\begin{align*} \frac {1}{a}\int \frac {dy}{y^{\frac {a-1}{a}}} & =\int dx\\ \frac {1}{a}\frac {ay}{y^{\frac {a-1}{a}}} & =x+c_{1}\\ \frac {y}{y^{\frac {a-1}{a}}} & =x+c_{1}\\ y^{1-\left ( \frac {a-1}{a}\right ) } & =x+c_{1}\\ y^{\frac {a-a+1}{a}} & =x+c_{1}\\ y^{\frac {1}{a}} & =x+c_{1}\\ y & =\left ( x+c_{1}\right ) ^{a}\end{align*}