Find the general solution of $\frac{{dy}}{{dx}} + ay = {e^{mx}}$.

Given differential equation is
$\frac{{dy}}{{dx}} + ay = {e^{mx}}$

which is a linear differential equation.

On comparing it with

$\frac{{dy}}{{dx}} + Py = Q,$ we get
$P = a,Q = {e^{mx}}$

${\rm{IF}} = {e^{\int P dx}} = {e^{\int a dx}} = {e^{ax}}$

The general solution is

$y \cdot {e^{ax}} = \int {{e^{mx}}} \cdot {e^{ax}}dx + C$

$\Rightarrow$ $y \cdot {e^{ax}} = \int {{e^{(m + a)x}}} dx + C$

$\Rightarrow$ $y \cdot {e^{ax}} = \frac{{{e^{(m + a)x}}}}{{(m + a)}} + C$

$\Rightarrow$ $(m + a)y = \frac{{{e^{(m + a)x}}}}{{{e^{ax}}}} + \frac{{(m + a)C}}{{{e^{ax}}}}$

$\Rightarrow$ $(m + a)y = {e^{mx}} + K{e^{ - ax}}$