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Math Help - Spring differential equation?

  1. #1
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    Spring differential equation?

    Motion of mass hanging from a spring is modeled y''=-π^2y
    Given Y(0)=5 and y(3/2)=8, find:
    The particular solution of this IVP and the first time (t is positive) at which the velocity is zero.

    I believe i've found the IVP and that it's y(t)=12.124sin(
    πt)+5cos(πt) .I'm not sure if it's correct though, but I don't know how to find the velocity. Thanks!
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Spring differential equation?

    Your solution is not correct. Using the fact that the roots of the associated auxiliary equation are r=\pm\pi i then the general solution is:

    y(t)=c_1\cos(\pi t)+c_2\sin(\pi t)

    We are given:

    y(0)=5 and y\left(\frac{3}{2} \right)=8

    Use these to determine the parameters (one of which you already have correct).

    To find the velocity, compute y'(t), then equate it to zero, and solve for the first positive value of t.
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  3. #3
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    Re: Spring differential equation?

    That's what I did for the solution. How am I not correct? I plugged in 0 for t and got that C2 was equal to 5 and then solved that when 3/2 was equal to t and 8 was equal to y(t) and got 12.124, can you help me get to the solution?
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  4. #4
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    Re: Spring differential equation?

    How did you get 12.124? At t = 3/2, the cosine function is 0 and the sine function is -1, so you should instead get...?
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  5. #5
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    Re: Spring differential equation?

    Oh, I plugged in 2/3. So, it should be -8. how would I solve for velocity given this info?
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  6. #6
    MHF Contributor MarkFL's Avatar
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    Re: Spring differential equation?

    Take the general solution:

    y(t)=5\cos(\pi t)-8\sin(\pi t)

    Compute the derivative to get the velocity function, equate it to zero, and use the smallest positive root.
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  7. #7
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    Re: Spring differential equation?

    0=-8πcost(πt)-5πsin(πt). This might sound really bad, but I dont know how to get the smallest positive root.
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  8. #8
    MHF Contributor MarkFL's Avatar
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    Re: Spring differential equation?

    I recommend dividing through by -\pi then using a linear combination identity.
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