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Math Help - Simple Demonstration Exercise

  1. #1
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    Simple Demonstration Exercise

    Please help!

    Given:

    y'(t) + a(t).y(t) = f(t), with "a" and "f" continuos in R
    a(t) >= c > 0
    lim (t->oo) f(t) = 0

    Demonstrate:

    Any solution y(t), verifies lim (t->oo) y(t) = 0

    Thank You.
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  2. #2
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    Re: Simple Demonstration Exercise

    Quote Originally Posted by Pedro View Post
    Please help!

    Given:

    y'(t) + a(t).y(t) = f(t), with "a" and "f" continuos in R
    a(t) >= c > 0
    lim (t->oo) f(t) = 0


    Demonstrate:

    Any solution y(t), verifies lim (t->oo) y(t) = 0

    Thank You.
    Here is a hint to get you started:

    The integrating factor for this equation is

    I(t)=e^{\int_{b}^{t}a(x)dx}

    Multiplying by this and writing the left hand side as a derivative gives

    \frac{d}{dt}\left(y(t)\cdot e^{\int_{b}^{t}a(x)dx}\right)=e^{\int_{b}^{t}a(x)d  x}f(t)

    Now use some of the assumptions to bound the solution to the equation.
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  3. #3
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    Re: Simple Demonstration Exercise

    Quote Originally Posted by TheEmptySet View Post

    \frac{d}{dt}\left(y(t)\cdot e^{\int_{b}^{t}a(x)dx}\right)=e^{\int_{b}^{t}a(x)d  x}f(t)

    Now use some of the assumptions to bound the solution to the equation.
    I've tried doing that. Maybe I'am missing something, but I'll get:

    Lim [ d/dt(y(t) * e^int{a(t).dt}) ] = 0,
    which gives that the integrand is constant,
    C = y(t) * e^int{a(t).dt}

    How can i reach that "y(t) = 0" when "t-->oo"?
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  4. #4
    Behold, the power of SARDINES!
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    Re: Simple Demonstration Exercise

    Quote Originally Posted by Pedro View Post
    I've tried doing that. Maybe I'am missing something, but I'll get:

    Lim [ d/dt(y(t) * e^int{a(t).dt}) ] = 0,
    which gives that the integrand is constant,
    C = y(t) * e^int{a(t).dt}

    How can i reach that "y(t) = 0" when "t-->oo"?
    How did you get that?

    If you integrate what I gave you above you should get

    y(t)\cdot e^{\int_{b}^{t}a(x)dx}=\int_{c}^{t}e^{\int_{b}^{y}  a(x)dx}f(y)dy

    Solving for y(t) gives

    y(t)=\frac{\int_{c}^{t}e^{\int_{b}^{y}a(x)dx}f(y)d  y}{e^{\int_{b}^{t}a(x)dx}}
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