Unable to find symbolic solution warning

% Define function names that I wish to solve 

syms x(t) y(t);

k1 = 0.02;
k2 = 0.00004;
k3 = 0.0004;
k4 = 0.04;

ode1 = diff(x) == k1*x-k2*x*y;
ode2 = diff(y) == k3*x*y-k4*y;

odes = [ode1;ode2];

S = dsolve(odes);

I keep getting a warning when I'm trying to solve this very simple system of odes.

Answer 1

[Star Strider]

It is nonllinear because of the ‘x*y’ terms, so it does not have an analytic solution that the Symbolic Math Toolbox can solve for.

Likely the only way to solve it is to use odeToVectorField and matlabFunction functions to create an anonymous function to use with the numeric ODE solvers, such as ode45 or ode15s:

[VF,Sbs] = odeToVectorField(odes);
odsefcn = matlabFunction(VF, 'Vars',{T,Y});

.

[Tinetendo Makata]

alright thanks!!

[Star Strider]

As always, my pleasurre!

[David Bloom]

@Star Strider

Mind giving me a hand since you seem quite adept at this?

I'm currently trying to solve a 2nd order ODE with dsolve and getting a similar error despite knowing that a solution does exist. Do you think I will need to take a similar approach?

ODE:

Initial Conditions:

: Where

My code:

syms r(t) r0

Dr = diff(r);

D2r = diff(r,2);

ode = D2r + (1/t)*Dr - r + r^3 == 0;

rSol(t,r0) = dsolve(ode,[Dr(0)==0,r(10)==0,r(0)==r0])

Warning: Unable to find symbolic solution.

 
rSol(t, r0) =
 
[ empty sym ]
 

[Walter Roberson]

This cannot work, for the same boundary condition reasons I recently happened to explain in https://www.mathworks.com/matlabcentral/answers/2058574-why-do-i-receive-the-error-while-running-the-code#answer_1368749

That said, MATLAB cannot handle it anyhow

syms r(t) r0

Dr = diff(r);

D2r = diff(r,2);

ode = D2r + (1/t)*Dr - r + r^3 == 0;
bc = [Dr(0)==0,r(10)==0,r(0)==r0];
rSol = dsolve(ode, bc)

Warning: Unable to find symbolic solution.

 
rSol =
 
[ empty sym ]
 

rSol = dsolve(ode)

Warning: Unable to find symbolic solution.

 
rSol =
 
[ empty sym ]
 

Maple and Mathematica cannot solve it either.

[David Bloom]

So how do I approach this problem then? In reality I have three initial conditions:

1)

2) positive value

3) which can be approximated as

I can live with choosing (2), but I'm not sure how to proceed with (1) and (3). Any thoughts?

[David Bloom]

I solved it myself and the issue comes down to the 1/r term. If you allow the interval to explicitly include r=0, then the symbolic solver fails to interpret it. However, if you change the interval from [0 20] to [1e-33 20] then it can indeed solve it.