Undetermined Coefficients: What happens when everything cancels?

An excellent question that I received in email today with regards to WeBWorK #9:

Hi professor Reitz, on problem number two for the new homework, when I try to solve for the particular solution, everything on the left side cancels.

This will happen when the expression on the right side of the equation also happens to be one of the solutions to the homogeneous equation.  We deal with it in much the same way we dealt with repeated roots in homogeneous equations: When guessing the particular solution to the nonhomogeneous equation, multiply your guess by (for example, use instead of .  Here’s an example.

Example: 

The general solution to the associated homogeneous equation is:

General solution:

Notice that one of the basic solutions involves , which matches the right hand side of the original equation.  Because of this, we would make the following guess for a particular solution:

Guess:

Notice that when you take the derivative, you will still end up with a term involving just (without the extra “t”), which will allow the left hand side of the equation to equal the on the right side.

Let me know if you have any questions (post a comment!),
Prof. Reitz