# Two functions in a differential equation

• Jun 13th 2013, 12:36 AM
rc101268
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.
• Jun 13th 2013, 11:25 PM
zzephod
Re: Two functions in a differential equation
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• Jun 14th 2013, 07:09 AM
HallsofIvy
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 $\displaystyle u= k_gl+ Ph^+$.
• Jul 6th 2013, 07:45 PM
zzephod
Re: Two functions in a differential equation
Quote:

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)

.
• Jul 7th 2013, 12:35 AM
JJacquelin
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.