Problem with the DSolve function

Question

OK guys, here's the thing, Mathematica does not return a solution for this system of differential equations:

DSolve[{p11'[t] == 2*p12[t] - (1/r)*p12[t]^2, 
p12'[t] == 
p22[t] - (w^2)*p11[t] - 2*z*w*p12[t] - (1/r)*p12[t]*p22[t], 
p22'[t] == -2*(w^2)*p12[t] - 
4*z*w*p22[t] - (1/r)*p22[t]^2 + (w^4)*q, p11[0] == 0, p12[0] == 0,
p22[0] == 0}, {p11[t], p12[t], p22[t]}, t]

Where z,w,r,q are parameters. Can anyone please try it on your system and tell me if you have the same problem?

I also tried with the NDSolve function

NDSolve[{p11'[t] == 2*p12[t] - (1/r)*p12[t]^2, 
p12'[t] == 
p22[t] - (w^2)*p11[t] - 2*z*w*p12[t] - (1/r)*p12[t]*p22[t], 
p22'[t] == -2*(w^2)*p12[t] - 
4*z*w*p22[t] - (1/r)*p22[t]^2 + (w^4)*q, p11[0] == 0, p12[0] == 0,
p22[0] == 0}, {p11[t], p12[t], p22[t]}, {t, 0, 10}]

but I get the error message:NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0.

I greatly appreciate your help

(I use mathematica 8.0 and of course VariationalMethods is turned on)

Answer 1

When using NDSolve all the parameters must have a numeric value.

solution[t_] = 
  With[{q = 1, r = 1, w = 1, z = 1}, 
   NDSolve[{p11'[t] == 2*p12[t] - (1/r)*p12[t]^2,
            p12'[t] == p22[t] - (w^2)*p11[t] - 2*z*w*p12[t] - (1/r)*p12[t] p22[t],
            p22'[t] == -2 (w^2)*p12[t] - 4*z*w*p22[t] - (1/r)*p22[t]^2 + (w^4)*q,
            p11[0] == 0,  p12[0] == 0, p22[0] == 0},
           {p11[t], p12[t], p22[t]}, {t, 0, 10}]][[1, All, 2]]

Plot[solution[t], {t, 0, 10}]

Answer 2

Just for fun, here is a version with Manipulate. So the values of parameters can be dynamically changed:

Manipulate[
 solution[t_] = 
  NDSolve[{p11'[t] == 2*p12[t] - (1/r)*p12[t]^2, 
     p12'[t] == 
      p22[t] - (w^2)*p11[t] - 2*z*w*p12[t] - (1/r)*p12[t] p22[t], 
     p22'[t] == -2 (w^2)*p12[t] - 
       4*z*w*p22[t] - (1/r)*p22[t]^2 + (w^4)*q, p11[0] == 0, 
     p12[0] == 0, p22[0] == 0}, {p11[t], p12[t], p22[t]}, {t, 0, 
     10}][[1, All, 2]];

 Plot[solution[t], {t, 0, 10}]
 , {{q, 1}, -10, 10, Appearance -> "Labeled"}
 , {{r, 1}, -10, 10, Appearance -> "Labeled"}
 , {{w, 1}, -10, 10, Appearance -> "Labeled"}
 , {{z, 1}, -10, 10, Appearance -> "Labeled"},
 TrackedSymbols :> {q, r, w, z}]