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Math Help - Seperable Differential Equation

  1. #1
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    Seperable Differential Equation

    dx/{dt}=-ax where x=x(t)

    x(t)=Ce^{-t} where do the constant C come from?

    dy/dt=ax-by where y=y(t) How do i find y(t)



    thanks for any help.
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  2. #2
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    Quote Originally Posted by charikaar View Post
    dx/{dt}=-ax where x=x(t)

    x(t)=Ce^{-t} where do the constant C come from?

    dy/dt=ax-by where y=y(t) How do i find y(t)



    thanks for any help.
    Dear charikaar,

    \frac{dx}/{dt}=-ax

    By seperation of variables,

    \frac{dx}{x}=-adt

    \int{\frac{dx}{x}}=-a\int{dt}

    lnx=-at+lnC where lnC is an arbirary constant.

    \frac{x}{C}=e^{-at}

    x=x(t)=Ce^{-at} ; there must be a typo, are you sure that the answer is x(t)=Ce^{-t}????

    Still thinking the second part......
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  3. #3
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    Quote Originally Posted by charikaar View Post
    dx/{dt}=-ax where x=x(t)

    x(t)=Ce^{-t} where do the constant C come from?

    dy/dt=ax-by where y=y(t) How do i find y(t)



    thanks for any help.
    Dear charikaar,

    \frac{dy}{dx}=ax-by

    \frac{dy}{dx}=aCe^{-at}-by by substituting the result of the first part....

    \frac{dy}{dx}+by=aCe^{-at}

    This is a linear differential equation. i.e: \frac{dy}{dx}+p(x)y=q(x) Therefore it could be solved by using an integrating factor.

    Can you do it from here??
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  4. #4
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    Nicely done.

    Thank you very much.
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