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Math Help - cant make sense of integral question

  1. #1
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    cant make sense of integral question

    I'm trying to make sense of this question.


    " Find the function x(t) where x(1) = 0 and examine the behavior of x(t) as t -->infinity.

    S(2x/2-x^2)dx = S dt " (S is the integral sign)
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by eniuqvw View Post
    I'm trying to make sense of this question.


    " Find the function x(t) where x(1) = 0 and examine the behavior of x(t) as t -->infinity.

    S(2x/2-x^2)dx = S dt " (S is the integral sign)
    \int\frac{2x}{2-x^2}\,dx=\int\,dt?

    Make a u sub: u=2-x^2\implies-\,du=2x\,dx.

    Thus, we have -\int\frac{\,du}{u}=\int\,dt\implies -\ln\left|u\right|=t+C\implies \ln\left|2-x^2\right|=-t+C \implies \left|2-x^2\right|=Ce^{-t}\implies 2-x^2=Ke^{-t}\implies x(t)=\sqrt{2-Ke^{-t}}

    Applying the initial condition, we have 0=\sqrt{2-Ke^{-1}}\implies K=2e.

    Therefore, x(t)=\sqrt{2-2e^{1-t}}.

    Now, what is \lim_{t\to\infty}\sqrt{2-2e^{1-t}}?
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