System of 4 ODEs solver

I'm solving a system of 4 ODEs but it wasn't working. It showed "Unable to find explicit solution". What should I do next?

Here is the ODE system:

syms V(t) 
syms X(t)
syms S(t) 
syms P(t)
Q = 30;
nuy_max = 0.2;
Ks = 0.5;
nuy = nuy_max * S / (Ks + S);
nuy_g = 0.05;
K = 0;
Y = 0.3;
S_F = 300;
ode1 = diff(V) == Q;
ode2 = diff(X) == (nuy - Q/V)*X;
ode3 = diff(P*V) == nuy_g * X * V - K*P*V;
ode4 = diff(S*V) == S_F * Q - nuy * X * V / Y;
odes = [ode1; ode2; ode3; ode4];
cond1=  V(0) == 200;
cond2 = S(0) == 1;
cond3 = X(0) == 20;
cond4 = P(0) == 0.1;
conds = [cond1; cond2 ; cond3; cond4];
Str = dsolve(odes, conds);
V.Sol(t) = Str.V
X.Sol(t) = Str.X
S.Sol(t) = Str.S
P.Sol(t) = Str.P

The result is

Please help me. Thanks in advance.

Answer 1

[Sam Chak]

Hi @Duong Quang Chi

If 'dsolve' cannot find explicit solutions for , , , and , we can use 'ode45' to numerically solve the ODEs and plot the results.

syms X(t) V(t) P(t) S(t) 
Q       = 30;
nuy_max = 0.2;
Ks      = 0.5;
nuy     = nuy_max*S/(Ks + S);
nuy_g   = 0.05;
K       = 0;
Y       = 0.3;
S_F     = 300;
eqn1    = diff(V)   == Q;
eqn2    = diff(X)   == (nuy - Q/V)*X;
eqn3    = diff(P*V) == nuy_g*X*V - K*P*V;
eqn4    = diff(S*V) == S_F*Q - nuy*X*V/Y;
eqns    = [eqn1 eqn2 eqn3 eqn4];
[V, S]  = odeToVectorField(eqns)

V = 

S = 

odefun  = matlabFunction(V, 'vars', {'t','Y'});
tspan   = [0 20];
X0      = 20;
V0      = 200;
P0      = 0.1;
S0      = 1;
y0      = [X0; V0; P0; S0];
[t, y]  = ode45(odefun, tspan, y0);

tb      = tiledlayout(2, 2, 'TileSpacing', 'Compact');
nexttile
plot(t, y(:,1)), grid on, title('X(t)')
nexttile
plot(t, y(:,2)), grid on, title('V(t)')
nexttile
plot(t, y(:,3)), grid on, title('P(t)')
nexttile
plot(t, y(:,4)), grid on, title('S(t)')
xlabel(tb, 'Time')

[Duong Quang Chi]

Omg it works :)) Though the explicit solutions cannot be found, these demonstration might be sufficient for my projects. Thanks a lot, I thought I was doomed.