How to solve a system of differential equations without symbolic math lab

I have three equations that are similar to these,

a, b, c, d, and e are all known and I have initial conditions for X, Y, and Z. I have equations saved in their own functions like this:

function dX = fun_dX(a,b,c,d,X,Y)
dX = a*X*(1-b*X)-c*X*Y-d
end

What functions/code should I be using to solve the system without the use of symbolic math lab? Also, how do I get it in a form that I can plot my answers all on one graph (using hold on)? I tried ode45 to solve them individually (for example solving the first equation with a given Y) but I believe the use of ode45 was wrong because the graph of X vs. t did not make sense.

Answer 1

[Walter Roberson]

a = whatever; b = whatever; c = whatever; d = whatever; e = whatever;

[t, xyz] = ode45(@(t, XYZ) odefun(t, XYZ, a, b, c, d, e), tspan, xyz0);
plot(t, xyz);

function dXYZ = odefun(t, XYZ, a, b, c, d, e)
   X = XYZ(1); Y = XYZ(2); Z = XYZ(3);
   dXYZ(1) = a*X*(1-b*X) - c*X*Y - d;
   dXYZ(2) = a + b*Y - d*X.^2*Y - c*X*Y;
   dXYZ(3) = e*Z + b^2*Z - c*Z*X _ d*Y*Z;
end

[Anna]

I'm getting a couple errors when I try to run this code,

Index exceeds the number of array elements (1).

Error in test2>odefun (line 8)
   X = XYZ(1); Y = XYZ(2); Z = XYZ(3);

Error in test2>@(t,XYZ)odefun(t,XYZ,a,b,c,d,e) (line 5)
[t, xyz] = ode45(@(t, XYZ) odefun(t, XYZ, a, b, c, d, e), tspan, xyz0);

Error in odearguments (line 90)
f0 = feval(ode,t0,y0,args{:});   % ODE15I sets args{1} to yp0.

Error in ode45 (line 115)
  odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

Error in test2 (line 5)
[t, xyz] = ode45(@(t, XYZ) odefun(t, XYZ, a, b, c, d, e), tspan, xyz0);

Also is it possible to set different initial values for X, Y, and Z?

[Anna]

I was able to get rid of some errors but there are new errors that I am having trouble understanding (I am fairly new to MATLAB). Test2 is the name of the file.

clc; clearvars;
a =5; b = 4; c = 2; d = 3; e = 1;
tspan = [0 100];
xyz0=[100 50 20];
[t, xyz] = ode45(@(t, XYZ) odefun(t, XYZ, a, b, c, d, e), tspan, xyz0);
plot(t, xyz)
function dXYZ = odefun(t, XYZ, a, b, c, d, e)
   X = XYZ(1); Y = XYZ(2); Z = XYZ(3);
   dXYZ(1) = a.*X.*(1-b.*X) - c.*X.*Y - d;
   dXYZ(2) = a + b.*Y - d.*X.^2.*Y - c.*X.*Y;
   dXYZ(3) = e.*Z + b.^2.*Z - c.*Z.*X + d.*Y.*Z;
end

Error using odearguments (line 93)
@(T,XYZ)ODEFUN(T,XYZ,A,B,C,D,E) must return a column vector.

Error in ode45 (line 115)
  odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

Error in test2 (line 5)
[t, xyz] = ode45(@(t, XYZ) odefun(t, XYZ, a, b, c, d, e), tspan, xyz0);

[Walter Roberson]

Instead of assigning to (1) and (2) and (3), assign to (1,1), (2,1) and (3,1)

[Anna]

What do you mean by that? I've tried changing xyz0 but it hasn't gotten rid of the errors.

[Walter Roberson]

function dXYZ = odefun(t, XYZ, a, b, c, d, e)
   X = XYZ(1); Y = XYZ(2); Z = XYZ(3);
   dXYZ(1,1) = a.*X.*(1-b.*X) - c.*X.*Y - d;
   dXYZ(2,1) = a + b.*Y - d.*X.^2.*Y - c.*X.*Y;
   dXYZ(3,1) = e.*Z + b.^2.*Z - c.*Z.*X + d.*Y.*Z;
end