Date: Tue, 16 Oct 2001 21:30:29 -0500
To: organism
Subject: help!!!!

Hey Jim,

I was wondering if you could give me a tip on these problems.  I keep 
trying to solve them but I get 0 for everything.  For my guess I tried y = 
(A(t^2) + Bt + C)e^t and I couldn't get it.  Can you please tell me if I am 
in the right track for number 15 on page 178.

Thank you,
A



Date: Tue, 16 Oct 2001 22:17:59 -0500 (CDT)
From: jr <organism>
Subject: Re: help!!!!

Problem 15 asks you to solve

y'' - 2y' + y = t e^t + 4

with some initial data.  I'm guessing you found the homogeneous solution:

yh = c1 e^t + c2 t e^t

For the particular solution, you need to notice a few things.  First the
RHS of the DE is a sum of two terms.  Find a particular solution for each
term, and then sum them to find the particular solution for the whole RHS.
Also, the first term looks like a first degree polynomial times e^t.
Therefore you might guess that (for that term alone):

yp = (At + B) e^t

However, those terms appear in the homogeneous solution in two places.
Thus the above guess won't work, and you need:

yp = t^2 (At + B) e^t

(For each appearance, bump up your guess by a factor of "t".)  For the "4"
term you might guess:

yp = A

and it'd work because there's no term like that in the homogeneous
solution.  (Beware this "A" is a different "A" than the one before.;)

Check out that summary on page 175 and the examples on the previous pages.
They might help some more.  Hope this helps!

j