3.3.4.10 Example 10 $y^{\prime }=a{y}^{\frac{a-1}{a}}$

$$y^{\prime }=a{y}^{\frac{a-1}{a}}$$

Integrating gives

$$\begin{array}{rl}\frac{1}{a}\int \frac{dy}{y^{\frac{a-1}{a}}}& =\int dx\\ \frac{1}{a}\frac{ay}{y^{\frac{a-1}{a}}}& =x+c_1\\ \frac{y}{y^{\frac{a-1}{a}}}& =x+c_1\\ y^{1-\left(\frac{a-1}{a}\right)}& =x+c_1\\ y^{\frac{a-a+1}{a}}& =x+c_1\\ y^{\frac{1}{a}}& =x+c_1\\ y& ={(x+c_1)}^a\end{array}$$