Solving a coupled set of differential equations in MATLAB?

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Kevin Bachovchin 2011-9-18

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https://ww2.mathworks.cn/matlabcentral/answers/16101-solving-a-coupled-set-of-differential-equations-in-matlab

评论： DEBASHIS PANDA 2015-9-11

Hello,

I have a set of coupled differential equations Ax + B dx/dt + C dy/dt = D Ey + F dy/dt + G dx/dt = H

The problem I have is that each derivative depends on the derivative of the other variable. Specifically dx/dt depends on dy/dt (not just merely y) and dy/dt also depends on dx/dt (not just merely x). Can I solve this system somehow using ode45? If not, is there some other differential equation solver I can use?

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Answer 1

Kevin Bachovchin 2012-1-13

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I was not able to find an easy way to do this. I had to define a new variable and use substitution in order to solve.

So originally my system of differential equations is Ax + B dx/dt + C dy/dt = D Ey + F dy/dt + G dx/dt = H

In matrix notation this can be represented as

[B C ; G F] [dx/dt ; dy/dt] = -[A 0 ; 0 E] [x ; y] + [D ; H]

Now we need to define new variables x_hat and y_hat

Let [x_hat ; y_hat] = [B C ; G F] [x ; y] So, in reverse, [x ; y] = inv([B C ; G F]) [x_hat ; y_hat]

Making this substitution into the differential equation system

[dx_hat/dt ; dy_hat/dt] = -[A 0 ; 0 E] inv([B C ; G F]) [x_hat ; y_hat] + [D ; H]

Now this system of differential equations can be solved for by ode45 in terms of x_hat and y_hat (since the derivatives of x_hat and y_hat do not depend on each other).

Of course, once you have the solution in terms of x_hat and y_hat, it's necessary to convert back to x and y.