How can I solve a system of differential equations where one equation contains the answer to another?

I have the following system of differential equations that shall be solved

where are constants, I am looking for a way to deal with the fact that dot and Ω dot are dependant on eachother. I have tried implementing the ode45 function but I did not get a logical result.

[ME]

You can 100% use MATLAB ODE solvers for this type of problem. Would you be able to upload your code so that we can see where you might have gone wrong?

Answer 1

[Star Strider]

If you have the Symbolic Math Toolbox, let it do the ‘heavy lifting’:

syms omegax(t) omegay(t) omegas Omega(t) epsilon sigma alphas t Y 
Eqns = [diff(omegax) == omegay*alphas + omegas*Omega*epsilon
        diff(omegay) == omegax*alphas - diff(Omega)*epsilon
        diff(Omega) == -diff(omegay) - Omega*sigma];

[VF,Subs] = odeToVectorField(Eqns)
Eqnsfcn = matlabFunction(VF, 'Vars',{t,Y,alphas,omegas,sigma,epsilon})

producing this system:

VF =
 -(alphas*Y[2] + epsilon*sigma*Y[3])/(epsilon - 1)
                 alphas*Y[1] + epsilon*omegas*Y[3]
          (alphas*Y[2] + sigma*Y[3])/(epsilon - 1)
 
 
Subs =
 omegay
 omegax
  Omega

and anonymous function to use with the numeric ODE solvers (slightly edited):

  Eqnsfcn =  @(t,Y,alphas,omegas,sigma,epsilon) [-(alphas.*Y(2)+epsilon.*sigma.*Y(3))./(epsilon-1.0);alphas.*Y(1)+epsilon.*omegas.*Y(3);(alphas.*Y(2)+sigma.*Y(3))./(epsilon-1.0)];

The ‘Subs’ output are the substitutions odeToVectorField made, with ‘omegay’ being ‘Y[1]’ ‘omegax’ being ‘Y[2]’, and ‘Omega’ being ‘Y[3]’.

[Erik Andersson]

Thank you, this has solved it.

[Star Strider]

As always, my pleasure!

When you do the numerical integration and plot the results, an easy way to use the legend function is:

legend(string(Subs))