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d^{2}y(t)/dt^{2}+ 5d^{2}y(t)/dt^{2}+ 4y(t) = 2e^{-2t}

Solved the complementary(homogenous) part and the function and got the roots of -1 and -4

so the y_{h}(t) is A_{1}.e^{-4t}+ A_{2}.e^{-t}

Forcing function is 2.e^{-2t}so y_{particular}(t) is A.e^{-2t}

Am I right here ? Or am I supposed to use Ate^{-2t}

Well, if I use the first one, the resultant function doesn't give me the 2.e^{-2t}when I put it into the differential equation, so there is something wrong obviously.

However F(t) or one of its derivatives are not identical to terms in the homogenous solution, so I think I have to use the first option, which is A.e^{-2t}

After proceeding I ended up with y_{p}(t) = 1/3.e^{-2t}

Initial values are y(0) = 0 and y^{(1)}(0) = 0

so, K_{1}= -1/9 and K_{2}= -2/9

Still couldn't find where I am wrong

Appreciate if you help me.

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# A problem with solution

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