Ordinary Differential Equations/Nonhomogeneous second order equations:Method of undetermined coefficients

Consider a differencial equation of the form

y''+p(t)y'+q(t)y=g(t)

Clerarly, this is not homogeneous, as $g(t)\neq 0$ .

So, to solve this, we first proceeed as normal, but assume that the equation is homogeneous; set $g(t)=0$ for now. Then the first part of the solution pans like

y(t)=c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t}.

.

Now we need to find the particular integral. To do this, make an appropriate substitution that relates to what $g(t)$ is. For instance, if $g(t)=e^{2x}$ , then take substitution $Ae^{2x}$ . As $y''$ and $y'$ are multiples of $y$ in this case, you'll simply get a linear equation in $A$ . Then just plug the value of $A$ in the equation.

Hence the solution is

y = general solution + particular integral.

There is one important caveat which you should be aware though. In the previous example for instance, if the general solution already had $e^{2x}$ , the substitution cannot be $Ae^{2x}$ , as the particular integral cannot be equal to the general solution. In such cases, you need to take the substitution as $Axe^{2x}$ .