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

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Solve the differential equation

 
Given that
 

Solution

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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