MATLAB: Using ode45 To Solve a System of 2nd Order ODEs

2nd order odesode'sode45system of ode's

I have the following set of 2 Second Order ODEs:

I need to solve these equations simultaneously, so that

in the first equation is used in

in the second equation.

I'm looking everywhere for the solution of my problem, but no luck.

I know from https://www.mathworks.com/help/matlab/math/choose-an-ode-solver.html that I need to transform a 2nd Order ODE in a 1st Order ODE, but how to do it with 2 ODEs and in an iterative way?

Can someone help me to use ode45 for this problem?

Thank you,

Rafael F.

Best Answer

Code them in the Symbolic Math Toolbox, and use the functions odeToVectorField and then matlabFunction to create an anonymous function that the ODE solvers such as ode45 can use to integrate them. There are other differential equation functions in the Symbolic Math Toolbox Mathematics documentation section if these turn out not to be appropriate for your system.

Related Solutions

MATLAB: System of Second order differential equations

See ode45(): https://www.mathworks.com/help/matlab/ref/ode45.html. Espically, check this example of 2nd order ODE using ode45(): https://www.mathworks.com/help/matlab/ref/ode45.html#bu3uj8b

MATLAB: Ode45

Yes, certainly possible. The basic approach would be to turn it into a system of first order odes of twice the dimension as follows (where x and xdot are vectors):

y1 = x
y2 = xdot

then, if A is invertible

y1' = y2
y2' = A \ (-B*y2 - c*y1 + p(t))

You'd then write a file to implement this in MATLAB like this

function dy = myode(t, y)
  n = numel(y)/2;
  y1 = y(1:n);
  y2 = y(n+1:2*n);
  dy = [y2; A \ (-B*y2 - c*y1 + p(t))
end

and solve it with ode45 (or whichever solver happens to be most suitable) as usual.

Best Answer

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MATLAB: System of Second order differential equations

MATLAB: Ode45

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