Question

Use the method of undetermined coefficients to find one solution of:

y'' + 2y' - 5y = (1t^2 + 8t -4)*exp(4t)

Note that the method finds a specific solution, not the general one.

y =?

Answer 1

For this, you'll want your guess to be the product of the two separate guesses. \[Ae^{4t} (Bt^2+Ct+D)\]

Answer 2

After plugging in the guess, gather the coefficients together and solve.

Answer 3

Thanks again. I appreciate it. Differential equations is just not my cup of tea. I wonder how that relates to Chemistry major though...

Answer 4

By the way, the expansion for that problem is (as I'm sure you've already seen) quite a monster, so I didn't particularly want to type it out here unless needed. I don't know much about Chemistry, but I do know that differential equations are important for things like the heat equation (which may have applications in Chemistry, I'm not sure).

Answer 5

I know. That's the impression I get from taking Physics (for Physics majors not engineers). My school has separate Physics classes for Biology, Chemistry/Physics/Geological Science majors, and one specifically for Engineers. I'm working out the problem right now and it's becoming really messay. Thanks again.

Answer 6

No problem

Answer 7

A hint to make this easier: You can ignore the exponential, find the particular solution for t^2 + 8t - 4, and then multiply that with the exponential.

Answer 8

Thanks for the hint. I tried expanding it and plugging it back into the equation. I think my arithmetic may be wrong since I got: A = 32, B = 1/32, C = 1/4 and D = -1/8.

Answer 9

Yeah, I definitely got something different. When solving \[y'' + 2y' - 5y = t^2 + 8t - 4\] I got \[2A + 4At + 2b - 5At^2 - 5Bt - 5C = t^2 + 8t - 4\] and thus the system of equations \[\begin{align*}2A + 2B - 5C &= -4 \\ 4A - 5B &= 8 \\ -5A &= 1\end{align*}\]

Answer 10

differential equations is probably one of the most useful maths in sciences

Answer 11

Hmm, I figured it was:
y =\[A \exp(4t)(Bt ^{2} + Ct + D)\]

I then took the 1st and 2nd derivative of that to plug into: y'' + 2y'' - 5y = exp(4t)(t^(2) + 8t -4). I think that's why our answers are different.

Answer 12

Oh, yeah. I used the hint I gave to make life a bit easier for myself. :) As long as it works when you plug it in, you should be fine.

Answer 13

When I did that and entered into webwork (online HW), my answer was wrong so I'll try your method and see what happens.

Answer 14

Just make sure you multiply my particular solution by e^(4t) after finding it