Cannot find solution with dsolve but I know that a relatively simple solution does exist

I'm using dsolve to solve a simple system of ODE's. I know there must be an analytical solution because our professor has tasked us with finding it, but MATLAB fails to produce a result.

Does anyone know why this would be failing?

Here is my code:

syms a(t) b(t) c(t) d(t) e(t)

ode1 = diff(a,t) == -5*a + .01*e;
ode2 = diff(b,t) == 5*a - 15*b;
ode3 = diff(c,t) == 5*a - .5*c*d;
ode4 = diff(d,t) == 15*b -.5*c*d;
ode5 = diff(e,t) == .5*c*d - .01*e;

odes = [ode1; ode2; ode3; ode4; ode5;];

conds = [a(0) == 5; b(0) == 0; c(0) == 0; d(0) == .5; e(0) == 1];

[a b c d e] = dsolve(odes, conds);

hold on
fplot(a)
fplot(b)
fplot(c)
fplot(d)
fplot(e)

Answer 1

[Star Strider]

The dsolve function apparently does not integrate nonlinear differential equations.

This is likely the best you can hope for:

syms a(t) b(t) c(t) d(t) e(t) t Y 
ode1 = diff(a,t) == -5*a + .01*e;
ode2 = diff(b,t) == 5*a - 15*b;
ode3 = diff(c,t) == 5*a - .5*c*d;
ode4 = diff(d,t) == 15*b -.5*c*d;
ode5 = diff(e,t) == .5*c*d - .01*e;
odes = [ode1; ode2; ode3; ode4; ode5;];
conds = [a(0) == 5; b(0) == 0; c(0) == 0; d(0) == .5; e(0) == 1];

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

tspan = [0 10];
ics = [0 5 0 0.5 1];
[t,y] = ode45(odes_fcn, tspan, ics);

figure
plot(t,y)
grid
legend(string(Subs), 'Location','best')

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