使用 Octave/Matlab 求解带有嵌入式非微分方程的微分方程组（见图）

Question

I have the following equation system (click to see picture) , and would like to solve for X(t), Y(t), Z(t), hopefully using Octave/Matlab, which I'm familiar with, but I would not mind solving it by any other means necessary.

Now, Fsolve is useful for regular equation systems, and Ode45, Lsode are useful for differential equations. But, what about this particular system? Note that the differential equation at the bottom contains not only Y, but, also, both X and Z, and those are dependent on the two non-differential equations above.

To be honest, I'm not quite sure how to approach a basic code to solve this system, and after spending some time thinking, I decided to ask for help. I really appreciate any guidance to solve this problem somewhat efficiently, if it is even possible. Almost any reply would be of some use to me right now.

Answer 1

If you know y , you can solve for x , and this even unconditionally as the second equation in monotonous in x

x = fsolve(@(x) y^2-1413.7*x-1095.2*cos(x)+2169, 0)

Then once you know x , you can solve for z using the known inverse cosine

z = acos(0.20978-cos(x))

This may actually fail to give a result if cos(x) comes close to -1 . One can artificially cut out that error, introducing a possibly wrong solution

z = acos(min(1,0.20978-cos(x)))

For simplicity, assemble these operations in a helper function

function [x,z] = solve_xz(y)
    x = fsolve(@(x) y^2-1413.7*x-1095.2*cos(x)+2169, 0);
    z = acos(min(1,0.20978-cos(x)));
end

Now use that to get the ODE for y

function dy = ode_y(t,y)
    [x,z] = solve_xz(y(1));
    dy = [ y(2); y(3); 6666.6667*(z-x)-333.3333*y(1)-33.3333*y(2)-5*y(3) ];
end

and apply the ODE solver of your choice. It is quite likely that the system is stiff, so ode45 might not be the best solver.