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I have this DE:

y'' + 2y' - 3y = 8e^{x}- 12e^{3x}

when I find homogeneous solution I get

y_{h}= c_{1}e^{x}+ c_{2}e^{-3x};

so now to find the particular solution by method of undetermined coefficients, do I set y to smth like this:

y = y_{1}+ y_{2}

where

y_{1}= Axe^{x},

y_{2}= A^{3x}?

since one of the solutions to auxiliary equation appears on the RHS of the DE and the other does not?

I don't need the full solution, just confirmation/correction of this part.

Thanks much!

EDIT: if I take the fact that if y1 + y2 is a solution, then y1 is a solution and y2 is a solution. I guess I answered my own question.

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# Undetermined coefficients

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