Ordinary Differential Equations/Nonhomogeneous second order equations:Method of undetermined coefficients
Consider a differencial equation of the form
Clerarly, this is not homogeneous, as .
So, to solve this, we first proceeed as normal, but assume that the equation is homogeneous; set for now. Then the first part of the solution pans like
Now we need to find the particular integral. To do this, make an appropriate substitution that relates to what is. For instance, if , then take substitution . As and are multiples of in this case, you'll simply get a linear equation in . Then just plug the value of 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 , the substitution cannot be , as the particular integral cannot be equal to the general solution. In such cases, you need to take the substitution as .
Example edit
Solve the differential equation
Given that
Solution edit
Take . Then
Hence the general form of the equation becomes
Then
. Then
Hence the final equation is