Question

Solve using undetermined coefficients

y'' + y = t(1 + sin t)

I got the t term to be At+B but the I'm kind of stuck on the tsint term.

Answer 1

Is the related equation supposed to be t(Ct+D)(cos t + sint t)?

Answer 2

You have to find the particular solution and the complementary solution and then use the thrm that says

\(y(t) = y_{c} + y_{p}\)

http://www.wolframalpha.com/input/?i=y%27%27+%2B+y+%3D+t%281+%2B+sin+t%29

Answer 3

I know that I need to find the homogeneous solution and the particular solution. I'm stuck at finding the relationship between y'' + y = t sin (t)

If it were just the sin term I could just use the e^t(A sin t + B cos t) (r=+/- i) relationship but I'm not sure about the t that's tacked on.

Would it just be (Ct+D)(E sin t + F cos t)?

Also, I don't actually need to evaluate the terms, just find the variable solution.

Answer 4

i don't think you can solve that using the method of undetermined coefficients.
use superposition, solve for t using undetermined coefficients. you can solve for the other using reduction of order or variation of parameters.

Answer 5

Hmm I'm looking at some-what similar equation right now (y'' + y = 3 sin 2t + t cos 2t) and they found the solution using undetermined coefficients but the relation they used was y = t e^(rt) instead Acost+Bsint.

Fixed a typo*

Answer 6

find this
y'' + y = t using method of undetermined coefficients.
the particular solution will be t
then find the particular solution of
y'' + y = t sin(t)

Answer 7

Yeah, that's what I did. I'm just having a hard time finding the relating equation to y'' + y = t sin(t)

Answer 8

interesting .. seems i didn't know this
http://en.wikipedia.org/wiki/Method_of_undetermined_coefficients#Examples

Answer 9

Oh damn, I should of checked wiki... haha.

Answer 10

seems that way ...

Answer 11

apparently I don't need to differentiate y_2 and find the complete particular solution. That'll save me like 20 min of working through the alg portion of the problem haha.

Answer 12

well ... this isn't quite nice method
http://upload.wikimedia.org/math/7/0/a/70a575f1c539021bd893575c05ff365d.png

Answer 13

seems both forms have the same solution type ... you can directly compare this result to t sin(t) and get the coefficients.

Answer 14

I actually didn't need to go through the algebraic portion of the problem haha. The only part that was needed was this = t (Ct+D)sin t + t (Et+F) cos t.

So the final solution for the problem is y = y1 + y2 = (At + B) + t[ (Ct+D)sin t + (Et+F) cos t ]

Answer 15

you need to determine the values of A, B, C, D ... the second order equation cannot have more than two coefficients.

Answer 16

I mean the problem actually only required the portion I typed out. The solution in the back of the book is actually the same except with A_0, A_1, B_0, B_1, etc. lol

The second part of the question asked that we use a CAS to determine the coefficients.

http://imageshack.us/a/img17/7984/asdasdasdwk.png

Answer 17

well i guess then it is.