First time question and never could format on these sites. Anyway, I have still a book from a course from years and years ago. Opening it up again to refresh some forgotten stuff, I try to solve a few initial value Laplace transformations but come up short each and every time. A small note in the margin indicates I didn't solve at least one of them back then either.
I can mention two, either one being shown how to solve would be amazing. I have the given answers from the book and will list them:
y''-2y' +2y = cos t , where y(0) = 1 & y'(0) = 0.
The case of such initial value problems equalling 0 are fine, but once I have like e.g. cos t, it gets messed up, no matter what I try.
The solution is meant to be: y= 1/5 (cos t - 2sin t 4e^tcos t - 2e^tsin t)
I get as usual to the partial fractions, but something seems to go wrong.
I need some hand holding to get there. I tried backtracking from the solution on a similar problem and so close yet so far away from solving it, I am at a total loss.
To show where I am at perhaps? : ( let f = Y(s) )
(s^2)f - s - 2sf + 2f = s(s^2 + 1)^-1
f[(s - 1)^2 + 1] = s(s^2 + 1)^-1 + s
From here on, I guess I am starting to mess things up, so possibly here is where I could need some injection of hand holding. Have been attempting this whole day not to mention probably a stressful day or two back at uni days :)
EDIT: To JM. I have done the mentioned fractional method before and failed; as I said, I need(ed) some hand holding not to not knowing what to do, but that failing to achieve the answer, whilst doing so. :)
But I probably have failed to do something right and am going to cross-check yet again. In fact, it was the choice of using As+B/Q1 + Cs+D/Q2 I thought possibly be wrong since I was unable to reach the conclusion.
Ya thanks JM, somehow doing it again, I did get the coefficients trivially.. sigh! :)) Thanks for hand holding!!
EDIT 2: Hang on, I am not free yet to finish, but I don't have the N/5th in there so before I accept, just gonna make sure it all works out! It did. Thought of showing all steps to help others but ok, I kinda got busy :) I hope I accepted answer correctly; am not used to this forum.