(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given a set of fundamental solutions {e^{x}*sinx*cosx, e^{x}*cos(2x)}

2. Relevant equations

y''+p(x)y'+q(x)=0

det W(y_{1},y_{2}) =Ce^{-∫p(x)dx}

3. The attempt at a solution

I took the determinant of the matrix to get

e^{2x}[cos(2x)cosxsinx-2sin(2x)sinxcosx-cos(2x)sinxcosx- cos^{2}xcos(2x)+sin^{2}xcos(2x)

Then using the identities sin^{2}x+cos^{2}x = 1 (for the last 2 terms) and sin(2x) = 2sinx*cosx (for the second term) and cancelling the 2 "cos(2x)cosxsinx" (first and third terms) I got

-e^{2x}(sin^{2}x+cos(2x))

Setting this equal to Ce^{-∫p(x)dx}and trying to solve I got as far as

ln(-e^{2x}(sin^{2}x+cos(2x))/C) = -∫p(x)dx

and now I'm a little bit stuck, I also don't know how to solve for q(x) here. Thanks for the help!

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# I Constructing a 2nd order homogenous DE given fundamental solution

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