MATLAB: How to indentify the variables when plotting after solving a coupled system of ODEs

differential equationsodeode45plotvariables

Hello, i had a system of coupled ODEs and i think i have solved it, the problem is that the variable Y which gives the output, i dont know how to tell what column is the response to the x(t), y(t), z(t) or their first derivatives. Help would be much appreciated. Here is the code
   syms x(t) y(t) z(t) p(t) Y 
p=sin((0.5+3.5*t)*t);
D1y = diff(y,t);
D2y = diff(y,t,2);
D1x = diff(x,t);
D2x = diff(x,t,2);
D1z = diff(z,t);
D2z = diff(z,t,2);
eqns=[D2x+2*x-1*y==p,D2y+(0.5*10*exp(-10*t))*D1y-x+2*y-z==0,D2z-y+2*z==0];
yode = odeToVectorField(eqns);
Yodefcn = matlabFunction(yode, 'Vars',[t Y]);
tspan = (0:0.001:10);
Y0 = [0 0,0 0,0 0];
[T,Y] = ode45(Yodefcn, tspan, Y0);
plot(T, Y)

Best Answer

The easiest way is to ask for a second output from your odeToVectorField call. It tells you the substitutions the function made to define each element in the returned column:
[yode,Sbs] = odeToVectorField(eqns);
here, the substitutions are:
Sbs =
  y
 Dy
  x
 Dx
  z
 Dz
The columns of ‘Yodefcn’ exactly reproduces those returned by odeToVectorField, and ode45 returns them in that order.
That can be used with some other assignments to automatically produce a legend that labels every line in the plot.
The Code —
syms x(t) y(t) z(t) p(t) Y
p=sin((0.5+3.5*t)*t);
D1y = diff(y,t);
D2y = diff(y,t,2);
D1x = diff(x,t);
D2x = diff(x,t,2);
D1z = diff(z,t);
D2z = diff(z,t,2);
eqns=[D2x+2*x-1*y==p,D2y+(0.5*10*exp(-10*t))*D1y-x+2*y-z==0,D2z-y+2*z==0];
[yode,Sbs] = odeToVectorField(eqns);
Yodefcn = matlabFunction(yode, 'Vars',[t Y]);
tspan = (0:0.001:10);
Y0 = [0 0,0 0,0 0];
[T,Y] = ode45(Yodefcn, tspan, Y0);
plot(T, Y)
Lgndc = sym2cell(Sbs);
Lgnds = regexp(sprintf('%s\n', Lgndc{:}), '\n', 'split');
legend(Lgnds(1:end-1), 'Location','N')
The Plot —
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