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 .


Solve the differential equation


Given that


Take  . Then


Hence the general form of the equation becomes

Now, the particular integral has to be found. To do so, we consider RHS:  . The substation then becomes  . Then   and  . Then the equation reduces to  . Hence  . The equation is now


 . Then


Hence the final equation is