# Thread: Which method is used to solve this ODE?

1. ## 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!

2. ## 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

3. ## 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.

4. ## 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.

5. ## Re: Which method is used to solve this ODE?

Hi,

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

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6. ## Re: Which method is used to solve this ODE?

Originally Posted by blaisem
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.