Example 3. Find the general solution of the equation $y^{\prime \prime}=2 y^{\prime}$.
Solution. The right-hand side of the given equation depends only on $y^{\prime}$. This is an equation of the form (10.9). Setting $y^{\prime}=p$, we find:

$$
\begin{gathered}
y^{\prime \prime}=\frac{d y^{\prime}}{d x}=\frac{d p}{d x}, \frac{d p}{d x}=2 y^{\prime}=2 p, \frac{d p}{p}=2 d x, \ln p= \\
=2 x+\ln C^{\prime}, p=e^{2 x+\ln C^{\prime}}=C^{\prime} e^{2 x}, \frac{d y}{d x}=C^{\prime} e^{2 x}, d y=C^{\prime} e^{2 x} d x \\
y=\frac{C^{\prime}}{2} e^{2 x}+C_{2}, y=C_{1} e^{2 x}+C_{2}
\end{gathered}
$$