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 Solve the differential equation Given that Solution 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