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im trying to solve the following equation using standard aux method:

d^2y/dx^2 + 3dy/dy +2y = cos x with conditions x(0)=-3 and x'(0) = 3

my aux eqn is:

ae^x + be^-2x

and my yp is;

a sin kx + b cos kx

i differentiate this twice and substitute into the original equation, and i get:

(-a sin x - b cos x)+3(a cos x-b sin x) + 2(a sin x+ b cos x) = cos x.

Now at this stage im not sure if this is right or not, can someone please confirm if this is right or not.

I ve done this question using the laplace method and the answer i get is;

-19/2e^x +32/5 e^2x + 1/10 cos x - 3/10 sin x.

Here my aux equation is not the same as what i get when i apply the laplace method.

Im not sure what is going wrong.

Someone help me please.

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# Im stuck

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