Particular integral for cos^2x term Watch

barnisaurusrex
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#1
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#1
What is the general form for the particular integral I need to use to solve the equation attached? The equation attached is the RHS of a second order ODE.
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the bear
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maybe {A \phi + B Cos \phi + C sin \phi + D Cos 2 \phi + E sin 2 \phi + F} .... because the cos squared term can be replaced with cos 2 \phi
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alan910_2
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Are you asking how to integrate that with respect to (phi)? cos^2(x) = 0.5(cos(2x) + 1)
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Ewan
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Assuming most of those are constants (except those depending explicitly on fi), you need to rearrange cos^2 into cos2x

Then use Acosx + Bsinx + Csin2x + Dcos2x + E afaik
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barnisaurusrex
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OK I've tried both methods, but don't get near my lecturer's solutions, which is attached. Any ideas? Note the equation is non-linear.
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barnisaurusrex
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Forgot to mention, everything is constant except for \phi
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matt2k8
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Getting rid of the power using the double angle formulae like people have said to do should lead to that :\ what's the LHS? I noticed the x sin x term in the PI, which suggests sin x is part of the complementary function, in which case the part of the PI to deal with the cos x term is of the form Ax cos x + Bx sin x (using x instead of phi as i cba with latex)
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barnisaurusrex
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u'' + u on the lhs. Using the double angle formula doesn't do it. How would you get the xsinx term? Does the form Axcosx + Bxsinx work? How would I know to use that form by looking at the first equation?
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matt2k8
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u`` + u gives the complementary function A cos x + B sin x, so any multiple of cos x and sin x in the LHS will give LHS = 0 so it'll all just cancel out if you try a PI of the form C cos x + D sin x. In general, if the PI you'd usually try is part of the CF, it'll all cancel out on the LHS meaning you end up with something inconsistent. But if you multiply it by x and try that instead it should usually work. (the reason I don't like this method of finding the PI much is all the "guessing" lol)
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Don John
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cos^2 (x) = 0.5[1+cos(2x)]
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the bear
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(Original post by Don John)
cos^2 (x) = 0.5[1+cos(2x)]
in the Holy Bible it says plainly that "Jesus wept".... raptors cannot weep so we must conclude that Our Lord was NOT a raptor.

the bear

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