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
editSolve the differential equation
Given that
Solution
editTake . 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