How do I solve a system of nonlinear differential equations like the one below?

As seen below (ode1 ode2 ode3) are my equations and c_1 to c_9 are just some constants which will be later determined. Is there any way to solve this without numerical methods? Thank you!

syms x(t) y(t) z(t);

c_1 = 1
c_2 = 2
c_3 = 1
c_4 = 1
c_5 = 1
c_6 = 1
c_7 = 1
c_8 = 1
c_9 = 1

ode1 = diff(x,t) == c_1*(c_3-x) + c_2*(x-y);
ode2 = diff(y,t) == c_4*(x-y) - c_5*c_6*y*(1-z) + c_7*c_6*exp(c_8 - c_9*z);
ode3 = diff(z,t) == c_5*y*(1-z) - exp(c_8 - c_9*z);
odes = [ode1; ode2; ode3]
cond1 = y(0) == 0;
cond2 = x(0) == 0;
cond3 = z(0) == 0;
conds = [cond1 cond2 cond3];

Answer 1

[Star Strider]

Add t and Y to the syms declaration, and add these to the end of the posted code:

[VF,Subs] = odeToVectorField(odes);
odefcn = matlabFunction(VF, 'Vars',{t,Y});

Then use ‘odefcn’ with the numerical ODE integrator of your choise (such as ode45) to integrate them numerically.

Use the ‘Subs’ variable to determine the variable assignment order in the function and in the outputs of the integration.

[Andrian Mirza]

It worked very well, thanks, how to plot the results though?

[Star Strider]

As always, my pleasure!

Try this —

syms x(t) y(t) z(t) t Y 
c_1 = 1

c_1 = 1

c_2 = 2

c_2 = 2

c_3 = 1

c_3 = 1

c_4 = 1

c_4 = 1

c_5 = 1

c_5 = 1

c_6 = 1

c_6 = 1

c_7 = 1

c_7 = 1

c_8 = 1

c_8 = 1

c_9 = 1

c_9 = 1

ode1 = diff(x,t) == c_1*(c_3-x) + c_2*(x-y);
ode2 = diff(y,t) == c_4*(x-y) - c_5*c_6*y*(1-z) + c_7*c_6*exp(c_8 - c_9*z);
ode3 = diff(z,t) == c_5*y*(1-z) - exp(c_8 - c_9*z);
odes = [ode1; ode2; ode3]

odes(t) = 

[VF,Subs] = odeToVectorField(odes)

VF = 

Subs = 

odefcn = matlabFunction(VF, 'Vars',{t,Y});

[t,y] = ode45(odefcn, [0 50], zeros(1,3)+1E-8);

figure
plot(t, y)
grid
legend(string(Subs), 'Location','best')
ylim([-1 1]*5)

.