You are not using the correct form for the particular solution, because the homogeneous solution is
Hence, you need to try a particular solution of the form:
Do you understand why?
Hi all,
I have been trying to solve this problem for hours and have looked at many examples, but I still can't solve it. It is a past exam question, so I don't know if there was an error or if I am just doing it wrong.
The problem says:
"Consider the ordinary differential equation
Using the Method of Undetermined Coefficients, find a particular solution of the inhomogeneous problem."
So I have tried to solve it using , but I can't get anything because the only result I get is , and no values for or . And would give instead of .
I would really appreciate it if someone could give me a hand. Thanks a lot!
You are not using the correct form for the particular solution, because the homogeneous solution is
Hence, you need to try a particular solution of the form:
Do you understand why?
No term in the particular solution can be a solution to the corresponding homogeneous solution, and so that's why the factor is necessary. Have you studied the annihilator method yet?
If we observe that:
we may then state that the differential operator:
annihilates
Now, applying this to the given ODE, we have:
Hence, the general solution must be of the form:
But, we see that the homogeneous solution to the original ODE is:
and we know the general solution is the superposition of the homogeneous and particular solutions:
Hence, we must conclude that the particular solution is of the form: