How to solve simultaneous differential equations?

% Parameters

mumax=0.2;

Ks=1.0;

Yxs=0.5;

Ypx=0.2;

Sf=10;

F=0.02;

Yps=1.0;

% Equaiotns

syms V(t) X(t) P(t) S(t)

Rg=mumax*(S/(S+Ks))*X;

Rp=Ypx*Rg;

% differential equations

ode1 = diff(V) == F;

ode2 = diff(X) == Rg-((X*F)/V);

ode3 = diff(P) == Rp-((P*F)/V);

ode4 = diff(S) == (F*(Sf-S)/V)-(Rg/Yxs)-(Rp/Yps);

cond1 = V(0) == 1;

cond2 = X(0) == 0.05;

cond3= P(0) == 0;

cond4 = S(0) == 10;

% Solution

sol1=dsolve(ode1,cond1);

sol2=dsolve(ode2,cond2);

sol3=dsolve(ode3,cond3);

sol4=dsolve(ode4,cond4)

This is my code, But sol4 is the message like "Warning: Symbolic solution not available." and dsolve 209 line.

What should I do fix the problem?

[Bjorn Gustavsson]

First: you have to call dsolve with all your coupled ODEs, see the dsolve-documentation for example of that.

Second: Not all differential equations have explicit analytical solutions - since X P and S are nonlinear in the solutions containing one factor of 1/V(t) this doesn't seem unlikely, it might be possible to find some cunning variable-transform that converts this into something with an analytical solution, that might be difficult. Therefore you might need to turn to numerical solutions, see the help and documentation for ode45, ode23, ode15s and their siblings.

Answer 1

[Star Strider]

It is nonlinear, so it does not have a symbolic solution.

Integrate it numerically instead —

% Parameters
mumax=0.2;
Ks=1.0;
Yxs=0.5;
Ypx=0.2;
Sf=10;
F=0.02;
Yps=1.0;
% Equaiotns
syms V(t) X(t) P(t) S(t) Y T 
Rg=mumax*(S/(S+Ks))*X;
Rp=Ypx*Rg;
% differential equations
ode1 = diff(V) == F;
ode2 = diff(X) == Rg-((X*F)/V);
ode3 = diff(P) == Rp-((P*F)/V);
ode4 = diff(S) == (F*(Sf-S)/V)-(Rg/Yxs)-(Rp/Yps);
cond1 = V(0) == 1;
cond2 = X(0) == 0.05;
cond3= P(0) == 0;
cond4 = S(0) == 10;
% Solution
[VF,Subs] = odeToVectorField([ode1, ode2,ode3,ode4])

VF = 

Subs = 

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

odefcn = function_handle with value:
    @(T,Y)[(Y(1).*(-1.0./5.0e+1))./Y(2)+(Y(1).*Y(4))./(Y(4).*5.0+5.0);1.0./5.0e+1;(Y(3).*(-1.0./5.0e+1))./Y(2)+(Y(1).*Y(4))./(Y(4).*2.5e+1+2.5e+1);-(Y(4)./5.0e+1-1.0./5.0)./Y(2)-(Y(1).*Y(4).*(1.1e+1./2.5e+1))./(Y(4)+1.0)]

[t,y] = ode89(odefcn, [0 100], [0.05 1 0 10]);

figure
plot(t, y)
legend(string(Subs))
grid

The ode89 function is new to R2021b. Use ode45 with earlier versions.

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[surrender]

Thx to perfect answer! :D

[Star Strider]

My pleasure!

If my Answer helped you solve your problem, please Accept it!

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