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anonymous

  • 5 years ago

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 =?

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

  2. apples
    • 5 years ago
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    After plugging in the guess, gather the coefficients together and solve.

  3. anonymous
    • 5 years ago
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    Thanks again. I appreciate it. Differential equations is just not my cup of tea. I wonder how that relates to Chemistry major though...

  4. apples
    • 5 years ago
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    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).

  5. anonymous
    • 5 years ago
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    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.

  6. apples
    • 5 years ago
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    No problem

  7. apples
    • 5 years ago
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    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.

  8. anonymous
    • 5 years ago
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    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.

  9. apples
    • 5 years ago
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    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 \[\begin{align*}2A + 2B - 5C &= -4 \\ 4A - 5B &= 8 \\ -5A &= 1\end{align*}\]

  10. anonymous
    • 5 years ago
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    differential equations is probably one of the most useful maths in sciences

  11. anonymous
    • 5 years ago
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    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.

  12. apples
    • 5 years ago
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    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.

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

  14. apples
    • 5 years ago
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    Just make sure you multiply my particular solution by e^(4t) after finding it

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