4.4.5.1 Example

$$2y^{\prime \prime }-e^y=0$$

Multiplying both sides by $y^{\prime }$ gives

$$2y^{\prime }{y}^{\prime \prime }-y^{\prime }{e}^y=0$$

Integrating

$$\begin{array}{rl}\int (2y^{\prime }{y}^{\prime \prime }-y^{\prime }{e}^y)dx& =c_1\\ {\left(y^{\prime }\right)}^2-e^y& =c_1\end{array}$$

Hence

$$y^{\prime }=\pm \sqrt{e^y+c_1}$$

Each of the above is separable, which are solved by integration.