DSolve doesn't know the answer to this differential-algebraic system

Questions
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1

I am having some trouble with solving a somewhat heavy differential equation system, which is consisted of 10 variables and other abstract parameters as follows:

A[t] == a0 + a1 * L[t]
B[t] == c0 + c1 * L[t]
M[t] == b0 + b1 * H[t]
Q[t] == (A[t] - 1) * B[t]
R[t] == x * q * A[t] * B[t] + f * M[t] 
F'[t] == d * (Q[t] - R[t])
G[t] == v * F[t]
H[t] == m0 + m1 * L[t] - m2 * G[t]
L[t] == p0 + p1 * J[t] - p2 * F[t]
J'[t] == n * (y - L[t])

I used DSolve as follows:

DSolve[{A[t] == a0 + a1 * L[t], B[t] == c0 + c1 * L[t], M[t] == b0 + b1 * H[t], Q[t] == (A[t] - 1) * B[t], 
R[t] == x * q * A[t] * B[t] + f * M[t], F'[t] == d * (Q[t] - R[t]), G[t] == v * F[t], H[t] == m0 + m1 * L[t] - m2 * G[t],
L[t] == p0 + p1 * J[t] - p2 * F[t], J'[t] == n * (y - L[t])}, {A[t], B[t], M[t], Q[t], R[t], F[t], G[t], H[t], L[t], J[t]}, t]

Mathematica can’t solve this and just return the same code. Am I missing something?

2

I tried to solve the problem. First converted A, B, M, Q, R, G, H, L to definitions, eg, A[t_]:= …, etc. Then get simultaneous DEs for F’[F[t],J[t]] and J’[F[t],J[t]]. DSolve couldn’t come up with a solution, even using Cloud Mathematica, in 30 minutes. I would suggest trying a numerical solution, after specifying initial conditions for F and J. Other possibilities: take Laplace transforms or represent F and J by series expansions.

3

Following S.M. Blinder’s approach one gets:

DSolve[
 {
  J'[t] == n (-p0 + y + p2 F[t] - p1 J[t]), 
  F'[t] == d ((-1 + a0 + a1 (p0 - p2 F[t] + p1 J[t])) (c0 + 
         c1 (p0 - p2 F[t] + p1 J[t])) - 
      q x (a0 + a1 (p0 - p2 F[t] + p1 J[t])) (c0 + 
         c1 (p0 - p2 F[t] + p1 J[t])) - 
      f (b0 + b1 (m0 - m2 v F[t] + 
            m1 (p0 - p2 F[t] + p1 J[t]))))}, {J[t], F[t]}, t]

which does indeed seem not to return a result after some waiting.

One can instead play with a restricted version of the problem where J[t] is a fixed function Jfixed[t] to see what sorts of answers one gets. Thus, for example, when Jfixed is linear this returns a result quickly involving Airy functions:

Jfixed[t] = t;

DSolve[
 {
  F'[t] == 
   d ((-1 + a0 + a1 (p0 - p2 F[t] + p1 Jfixed[t])) (c0 + 
         c1 (p0 - p2 F[t] + p1 Jfixed[t])) - 
      q x (a0 + a1 (p0 - p2 F[t] + p1 Jfixed[t])) (c0 + 
         c1 (p0 - p2 F[t] + p1 Jfixed[t])) - 
      f (b0 + b1 (m0 - m2 v F[t] + 
            m1 (p0 - p2 F[t] + p1 Jfixed[t]))))}, {F[t]}, t]

For Jfixed[t] = t and Jfixed[t] = t^2 a solution is returned. For higher integer exponents it appears that no solution is returned (i.e., DSolve eventually returns unevaluated).

Experimenting with different functions for Jfixed[t] can give some insight into the solvability of your more complicated system when J[t] itself has a differential equation to govern its behavior coupled with F[t].

Of course in the Jfixed[t] case you could use the expression for F’[t] to develop a power series expansion for F[t]. And in the J[t] case you might be able to use the two coupled differential equations to develop a recurrence relation that governs the interconnected power series expansions of the two functions.

4

Hi S.M.Blinder, Thank you so much for your helpful comments. I have been struggling last couple of days with trying to implement your suggestions. I have one followup equation if you allow.

Actually, there is one more equation in the above model which is:

Z'[t] = Q'[t] - R'[t]

In this case, do you think it is mathematically possible to reduce the model into a system of two differential equations in Z and L:

Z'[Z[t],L[t]]
L'[Z[t],L[t]]

Maybe this is more of a math question.

Any comments will be greatly appreciated!

5

Hi David, I really appreciate for your great comments. I’m not sure if Jfixed[t] approach will be appropriate for the specification of my model.

Anyway, I posted a followup question to the person to whom you made a reply. If you could take a look at and maybe could have some comments, it will be greatly appreciated!

Thank you so much!

ppp