Somewhat Easy Differential Equation Question

[bosox09]

This is a straightforward concept-type question that I already know the answer to, but I need someone to shed some light on HOW this is figured out. I have an idea but this is probably going to be asked of me on my final exam and I want to know the method behind this problem.

Homework Statement

Given the equation 5y''(t)−y'(t)+7y(t) = 3te^4tcos2t + t²e^t + 4t³e^2tsin4t − (2/3)e^t + 9e^2tcos4t, if the Method of Undetermined Coefficients and the Principle of Superposition are used to find a particular solution, what is the minimum number of nonhomogeneous equations which must be solved?

Homework Equations

The answer is 3. I believe this is because each individual, unique term on the right side has to be solved separately, and then each of these solutions are summed to find the general solution to the diff eq. Even though there are 5 separate terms on the right side, the e^t terms can be grouped and solved at once, and since cost + sint = 1, I'm guessing the sin4t and cos4t terms can be grouped as well. Am I way off?

Thanks for the help, hopefully this is a quickie.

 

[AEM]

Don't you mean $$cos^2(t) + sin^2(t) = 1$$?

 

[bosox09]

Hahahahahaha ugh...duh! Alright then can anyone explain why it's only 3 instead of 4?