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Math Help - Which method is used to solve this ODE?

  1. #1
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    Which method is used to solve this ODE?

    Hi, I am trying to solve an assignment in chemistry which involves an ODE. The equation is:

    \frac{d[I]}{dt}+k_{b}[I]=k_{a}[A]_{0}e^{-k_{a}(t)}

    where ka, kb, and [A]0 are all constants.

    My book provides a solution for equations of the form:

    \frac{df}{dx}+af=b

    as

    fe^{\int{adx}}=\int{e^{\int{adx}}bdx}+C

    I plugged this in for my case and the correct answer came out. For future reference, what method can I look up to solve an equation like this, or how is it actually classified? I think it is nonlinear?

    Thank you!
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  2. #2
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    Re: Which method is used to solve this ODE?

    the method used is to find what's known as an integrating factor. Integrating factor - Wikipedia, the free encyclopedia
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  3. #3
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    Re: Which method is used to solve this ODE?

    Another method: since this is a linear equation, you can separate it into a "homogeneous" part, \frac{d[I]}{dt}+ k_b[I]= 0 which is easy to solve since it is "separable": \frac{d[I]}{I}= -k_b dt. Find the general solution to that and add any one solution to the entire equation. And since the equation has constant coefficients, a function of the form Ae^{-k_at} should work. Put that into the equation and see what A must be in order that this must satisfy the equation.
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    Re: Which method is used to solve this ODE?

    Thanks for the help to both of you!

    One question: HallsofIvy mentions that the equation is separable into [tex]\frac{d[I]}{I}=-k_{b}dt[\tex]

    but I thought it wasn't separable because of the exponential term on the right-hand side of the original equation? Can you just ignore that term, since it doesn't seem to appear in HallsofIvy's homogeneous equation?

    I also wasn't sure if it was linear because (at least in Wikipedia) a linear equation has all unknown functions and their derivatives to the first power, but I had a function f(t) that was an exponential function.
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  5. #5
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    Re: Which method is used to solve this ODE?

    Hi,

    Integrating factor will solve this. Try this out at once.

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    Re: Which method is used to solve this ODE?

    Quote Originally Posted by blaisem View Post
    Thanks for the help to both of you!

    One question: HallsofIvy mentions that the equation is separable into [tex]\frac{d[I]}{I}=-k_{b}dt[\tex]

    but I thought it wasn't separable because of the exponential term on the right-hand side of the original equation? Can you just ignore that term, since it doesn't seem to appear in HallsofIvy's homogeneous equation?

    I also wasn't sure if it was linear because (at least in Wikipedia) a linear equation has all unknown functions and their derivatives to the first power, but I had a function f(t) that was an exponential function.
    I did NOT say that the original equation was "separable". I said that the "associated homogeneous equation, \frac{dI}{dt}+ k_bI=  0
    was separable.

    When Wikipedia refers to "unknown functions" here, it is referring to the function, I, that you are trying to solve for. Any other function is a "known" function.
    Thanks from blaisem
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