Question

Determine a suitable form for Y(t) if the method of undetermined coefficients is to be
used and Find a particular solution

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

Answer 1

I'm sorry, but that's "y" to the what?

Answer 2

It's a second derivative @ChrisTopher987654321

Answer 3

@OOOPS That would be fine if we were doing variation of parameters, and it can be helpful to sort of think about that to get a hint towards doing it this way.

Generally when you see a sine or cosine, you're going to want to pick something that has both because they are each others derivatives. Here is no exception. It looks like since it ends up being:

t+tsint

that you should pick something like

t+tsint+tcost

and now just slap on some coefficients

y=At+Btsint+Ctcost

Now try to solve and see what happens. If you try something and it doesn't work, whatever extra stuff pops out is sort of your hint as to what to add in. If you just ran through with just y=At+Btsint you would have gotten some extra cosine terms and no way to get rid of them, but I'm saving you the trouble; you're free to try it out for yourself though so don't take my word for it!

Answer 4

@Kainui the question is about "using undetermined coefficient method"

Answer 5

Am I not doing undetermined coefficients? @_@

Answer 6

If you just apply directly like what you do above without finding the homogeneous part, you make a mistake with repeating solution. As I posted and deleted, (hihihi) the homogeneous part has solution : y = A cos(t) + Bsin(t) , so that sin(t), and cos(t) cannot be in the non-homogenous part. They must be *t to get t^2....., am I right?

Answer 7

You're probably right, I never really do the undetermined coefficients method since variation of parameters is usually an option in these cases.