Solving a matrix valued ODE using dsolve in Matlab?

[Terp]

Hi all. I've a MATLAB assignment and I'm completely lost on one of the questions simply becuase I can't figure out the MATLAB syntax to do the problem correctly. The equation is a simple x' = Ax problem, where A is a 2x2 matrix, which I have to solve using dsolve. I have no problem doing by hand, but I can't figure it out in matlab. My book says "The solutions to systems of linear equations can also be found with a direct application of dsolve," but goes into no further explanation (this book is really worthless). Any help is appreciated...thanks! :)

-Brady

 

[Aaron Bex]

Here is an example of how to solve a system of linear equations using dsolve in MATLAB.

Suppose we have the following system of linear equations:

x' = Ax

where A is a 2x2 matrix given by:

A = [2 3; 1 2]

We can solve this system with dsolve as follows:

syms x1(t) x2(t)
eqns = [diff(x1(t)) == 2*x1(t) + 3*x2(t), diff(x2(t)) == x1(t) + 2*x2(t)];
[x1Sol(t), x2Sol(t)] = dsolve(eqns);

The output will be two symbolic solutions, one for x1(t) and one for x2(t).

Hope this helps!

 

[tacman]

Hi Brady,

Solving a matrix valued ODE using dsolve in Matlab can be done by following a few simple steps. First, you need to define your matrix A and your initial conditions for x. Next, you can use the dsolve function by passing in your equation, initial conditions, and the variable you want to solve for (in this case, x). The syntax will look something like this:

A = [a b; c d]; % define your matrix A
x0 = [x0; y0]; % define your initial conditions for x
syms x(t) % declare x as a symbolic variable
eqn = diff(x,t) == A*x; % create your ODE equation
sol = dsolve(eqn, x(0) == x0); % solve the ODE using dsolve

The solution will be in the form of a symbolic expression, so you will need to use the subs function to substitute in your values for a numerical solution. For example:

x_sol = subs(sol, [a b c d x0 y0], [1 2 3 4 5 6]); % substitute in values for a, b, c, d, x0, and y0
x_sol = x_sol(1); % extract the first element of the solution vector
x_sol = matlabFunction(x_sol); % convert the symbolic expression to a function
x_sol = x_sol(t); % evaluate the function at different values of t

I hope this helps you solve your problem in Matlab. If you need further assistance, feel free to reach out for more clarification. Good luck with your assignment!