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Math Help - general solution of differential equations

  1. #1
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    general solution of differential equations

    I've to find the general solution and determine how the solutions behave as t-->infinity.

    1. y'-2y=t^(2)e^(2t)
    μy'-2μy=μt^2e^(2t)
    d/dt (μ(t)y(t)) = μy'+μ'y
    What we need: μ' = -2μ, which is a separable equation with μ=e^(-2t)
    d/dx(e^(-2t)y)=e^(-2t)t^2e^(2t)=t^2
    Integrate: e^(-2t)y=t^3/3+C
    Did I do this right and how does it behave?

    2. y'+y= te^(-t)+1. I couldn't figure out this one 8(
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  2. #2
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    Quote Originally Posted by Taurus3 View Post
    I've to find the general solution and determine how the solutions behave as t-->infinity.

    1. y'-2y=t^(2)e^(2t)
    I don't think this is separable.

    Find and integrating factor \displaystyle I =e^{\int -2~dt}

    What do you get?
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  3. #3
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    For 1 you are correct. Now solve for \displaystyle y in terms of \displaystyle t and see what happens as you make \displaystyle t \to \infty.

    For 2, the integrating factor is \displaystyle e^{\int{1\,dt}} = e^t. Multiply both sides by \displaystyle e^t and see what you can do...
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  4. #4
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    ok, so for the 2nd question:
    p(t)=1, g(t)=te^(-t)+1
    μ(t)=e^t
    So integral of μ(t)g(t) = integral of e^t(x-e^x(x+1))
    Do I have it right so far?
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  5. #5
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    Your working out is a bit all over the place.

    You should have

    \displaystyle e^t\frac{dy}{dt} + e^ty = e^t(t\,e^{-t} + 1)

    \displaystyle \frac{d}{dt}\left(e^ty\right) = t + e^t

    \displaystyle e^ty = \int{t + e^t\,dt}.

    Go from here.
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  6. #6
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    so the final answer for 1 is: y=(t^3/3+C)e^(2t)
    2: y=(e^t+t^2/2+C)/e^t right?
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  7. #7
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    Correct. Now see what happens as you make \displaystyle t \to \infty.
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  8. #8
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    1. The If c is non negative, then the solution grows exponentially large in magnitude. So the solutions diverge as t gets bigger. The boundary between solutions that ultimately grow negatively occurs when c is negative.

    2. uggh...can't figure this one out.
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  9. #9
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    2. Why don't you just divide through by the exponential, term-by-term, and see what happens to each term as t goes to infinity?
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  10. #10
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    nvm. I got it haha.
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  11. #11
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    Quote Originally Posted by Taurus3 View Post
    nvm. I got it haha.
    Great!
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