# Thread: Two functions in a differential equation

1. ## Two functions in a differential equation

Hi All
I am trying to solve equation 12 in the picture in order to get the function for h+, this is necessary to compute a model.
The problem is e- is also a function but it has its own equation, e- and h+ change with time t. All other parameters are constants.
Thanks.

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3. ## Re: Two functions in a differential equation

Your first equation is $\displaystyle \frac{dh^+}{dt}= k_gl+ k_r(h^+)(e^-)- k_l(h^+)n_A\Omega_A= (k_re^-- k_ln_A\Omega_A)h^+$

You can simplify that by letting $\displaystyle P= k_re^-- k ln_A\Omega_A$ so that the equation is just [tex]\frac{dh^+}{dt}= k_gl+ Ph^+[tex].

Separate that as $\displaystyle \frac{dh^+}{k_gl+ Ph^+}= dt$ and integrate both sides. You can easily integrate on the left using the substitution $\displaystyle u= k_gl+ Ph^+$.

4. ## Re: Two functions in a differential equation

Originally Posted by HallsofIvy
Your first equation is $\displaystyle \frac{dh^+}{dt}= k_gl+ k_r(h^+)(e^-)- k_l(h^+)n_A\Omega_A= (k_re^-- k_ln_A\Omega_A)h^+$

You can simplify that by letting $\displaystyle P= k_re^-- k ln_A\Omega_A$ so that the equation is just $\displaystyle \frac{dh^+}{dt}= k_gl+ Ph^+$.

Separate that as $\displaystyle \frac{dh^+}{k_gl+ Ph^+}= dt$ and integrate both sides. You can easily integrate on the left using the substitution $\displaystyle u= k_gl+ Ph^+$.
You have not separated the variables since your $\displaystyle P$ is a function of $\displaystyle e^-$ which is a function of time.

(this is a pair of simultaneous non-linear coupled first order ordinary differential equations)

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5. ## Re: Two functions in a differential equation

Hi !

The couple of non-linear ODEs can be transformed to one non-linear ODE with only one unknown function. (see attachment)
But this kind of non-linear ODE is too complicated to be soved analytically.
So, don't expect to find an analytical solution. Try to simplify the problem in using some appoximations on some restricted ranges of the variables (for example by remplacing h*e by a linear function of h an e). If no approximate linearization is possible, you will probably have to use a numerical method of solving.