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 .

ExampleEdit

Solve the differential equation

 

Given that
 

SolutionEdit

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  
 . Then

 

Hence the final equation is