Hello everyone
I'm a bit stuck on an FP2 second order differential equation - any help would be really appreciated!
The equation I have to solve is:
dt2d2x+2dtdx+5x=10sintand the question says: 'Find the values of p and q so that
x=psint+qcost is a solution of the differential equation.'
I got the complementary function to be:
x=e−t(Acos2t+Bsin2t) which I'm fairly sure is okay.
The first thing I'm stuck on is whether to use the particular integral of:
x=(λcosωt+μsinωt)or to multiply that by t, because the form has already been used in the complementary function (sort of)... I don't really know what I'm talking about.
I tried it both ways - with the extra t, I ended up with
λ and
μ in terms of t, when in all the examples I've seen it should cancel out. Without the extra t, I got
λ=−1 and
μ=2.
My next problem is getting the answer in the form
x=psint+qcost. If what I have done so far is right, my general solution is:
x=e−t(Acos2t+Bsin2t)−cost+2sint...which isn't in that form. (I've tried simplifying it down but it gets pretty messy because of the double angles and I'm not sure it would even work out.)
Could someone point me in the right direction please?
(Sorry if I'm being completely dim!
)