Differential Equations. My issue is I don\'t know how to craft a undetermined co

Question

Differential Equations. My issue is I don't know how to craft a undetermined co-efficient guess using method of determinants. I know how to do it with implicet differentiation but not with matrixes/vectors.
When I have a tail vector like:
|3*(e^2t)|
|t*(e^2t)|
What's my guess for the y-section of the vector? Is it
|A(e^2t) |
|Bt*(e^2t) + Ce^2t|
in the vector?
Here's the problem. Solve:

x'=2x + y + 3e^2t
y'=-4x+2y+te^2t

I've already found the homogeous solution which involves imaginary numbers. Whats the guess look like for the tail segments of x' and y' vector?

Explanation / Answer

Since homogeneous soution involves imaginary values then your solution will be in terms of Cosine and sine function.

And it has nothing to do with the next part of the solution.

So for 3e2t Assume it to be Ae2t and for te2t assume it to be ( Bt + C) e2t

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