# Solving system of ODE's contains complex equations

3 views (last 30 days)
Bathala Teja on 30 Aug 2021
Answered: Bjorn Gustavsson on 31 Aug 2021
Iam forming system of 3 ODE's(dy1/dt, dy2/dt, dy3/dt) using matrices A, B and V.
A and B matrices(both 3*3) are interms of some variable teta.
I want to substitute y3 in place of teta and proceed solving ODE's.
My ODE is inv(B).*(V-(A*Y)), where V = column matrix of order 3.
i tried like this
syms teta
Ai = cos(teta).*([2 5 1;
7 9 0;
4 1 3])
Bi = sin(teta).*([1 2 3;
4 5 6;
7 8 9])
V = ones(3, 1)
odefun = @(t,Y) inv(B).*(V-(A*Y));
A = subs(Ai, teta, Y(3))
B = subs(Bi, teta, Y(3))
tspan = [0 10];
Y0=zeros(3,1);
[t, Y] = ode45(odefun, tspan, Y0);
Y1 = Y(:, 1);
hold on
plot(t, Y1);
Y2 = Y(:, 1);
plot(t, Y2);
Y3 = Y(:, 1);
plot(t, Y3);
legend('Y1','Y2','Y3')
But iam getting this error
Unrecognized function or variable 'Y'.
Error in Untitled5 (line 43)
A = subs(Ai, teta, Y(3))
Facing issue in substituting Y3, how to resolve that?
Bathala Teja on 31 Aug 2021
I can avoid this by changing intial conditions.
My main question is how to solve this kind of problem. Dont consider values seriously.

Bjorn Gustavsson on 31 Aug 2021
The way it looks to me you have a problem something like this: One problem is that the B-matrix you've given is singular which might be a problem. For problems like this there's not much need to doodle on with symbolic calculations, just implement the mass-matrix and ODE-function as matlab-functions and look for a numeric solution with ode45 or ode15s etc. Look at the help and documentation for those functions and focus in particular on the examples with mass-matrices. I'd figure the ODE-function would look something like:
function dydt = yourODE(t,y,A,V)
dydt = V - cos(y(3))*A*Y(:);
end
and the mass-matrix-function:
function M = yourMassM(t,y,B)
M = sin(y(3))*B;
end
Then you'll have to inform the ode-solver about the mass-matrix:
options = odeset('Mass',@(t,y) yourMassM(t,Y,B));
and then solve with some suitable initial conditions.
y0 = [pi/7,pi/11,pi/13];
t_span = [0,exp(17*pi)];
sol = ode15s(@(t,y) yourODE(t,y,A,V)),t_span,y0,options);
Since your mass-matrix is singular I dont guarantee that this works.
HTH

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