Solve differential equation from Matlab

Hello guys,

I want to ask about, how to solve the differential equation in MATLAB for the equation:

dx/dt = (ka*C(xmax-x)-kd*x)/(1+ka*x)

thx

[Jan]

You solve this using an integration. Do you want to solve this numerically (as initial value problem) or symbolically (hoping that there is a closed form equation for the integral)?

[Sabella Huang]

Hello, thx for your suggestions. I want to solve this just symbolically

[John D'Errico]

Are ka and kd constants, or is k separate from a and d, so there are three unknown constants. Is xmax known? Is it a value that x may not exceed? Is C a function? Or is C just a known constant?

When you are too vague, you make it impossible for people to help you.

Answer 1

[John D'Errico]

Making various assumptions, we see the solution as:

syms ka kd C xmax
syms x(t)
dsolve(diff(x) == (ka*C*(xmax-x)-kd*x)/(1+ka*x))

ans = 

Where W0 is the Lambert W function. Actually, the zero'th branch thereof, so lambertw(0,u).

help lambertw

 LAMBERTW Lambert's W function.
    W = LAMBERTW(Z) solves W*exp(W) = Z.
    W = LAMBERTW(K,Z) is the K-th branch of this multi-valued function.
 
    References:
    [1] Robert M. Corless, G. H. Gonnet, D. E. G. Hare,
    D. J. Jeffrey, and D. E. Knuth, "On the Lambert W Function",
    Advances in Computational Mathematics, volume 5, 1996, pp. 329-359.
 
    [2] Corless, Jeffrey, Knuth, "A Sequence of Series
    for The Lambert W Function", ISSAC '97

    Documentation for lambertw
       doc lambertw

    Other functions named lambertw

       sym/lambertw

[Walter Roberson]

note that is two different solutions

[Sabella Huang]

ok this work... thx

Answer 2

[Bjorn Gustavsson]

For a numerical integration of this ODE look at the help and documentation for ode45 and its siblings, also look at the examples and demos related to these ODE-solvers, you find these in odeexamples. That is perhaps the best way to get started (maybe it will not make you an expert in DE in general, but it will get you going.) You can also read the code of the different demos and adapt them to suit your problem.

For an analytical solution you might have success with dsolve from the symbolic toolbox if you have access to that.

HTH