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Thread: 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
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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 $\displaystyle r=\pm\pi i$ then the general solution is:

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

    We are given:

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

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

    To find the velocity, compute $\displaystyle y'(t)$, then equate it to zero, and solve for the first positive value of $\displaystyle t$.
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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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    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:

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

    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 $\displaystyle -\pi$ then using a linear combination identity.
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