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

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Erik Andersson
Erik Andersson 2019 年 11 月 22 日
Commented: Star Strider 2019 年 11 月 22 日
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.

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ME
ME 2019 年 11 月 22 日
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?

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Star Strider
Star Strider 2019 年 11 月 22 日
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]’.

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Star Strider
Star Strider 2019 年 11 月 22 日
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))

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