# Mathematica doesn't know the answer to this differential equation system

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]


When run, the result I get is just a repetition of the same code. Any comments will be greatly appreciated!

• Mma returns the same code, if it does not know the solution. Commented Oct 2, 2014 at 8:06
• @AlexeiBoulbitch Thanks. Then there is no way to solve this kind of heavy DEsystem?
– ppp
Commented Oct 2, 2014 at 12:48
• Depends upon your goal. May be you could simplify it a bit, to help Mma. You may think about solving it numerically. Then you need to somehow fix the values of the constants. Commented Oct 2, 2014 at 13:25

 expr1 = MapAt[
Collect[Expand[#], {F[t], F[t]^2, J[t],
J[t]^2}] &, (F'[t] == d*(Q[t] - R[t]) /.
R[t] -> x*q*A[t]*B[t] + f*M[t]) /. M[t] -> b0 + b1*H[t] /.
Q[t] -> (A[t] - 1)*B[t] /. H[t] -> m0 + m1*L[t] - m2*G[t] /.
G[t] -> v*F[t] /. {A[t] -> a0 + a1*L[t],
B[t] -> c0 + c1*L[t]} /. L[t] -> p0 + p1*J[t] - p2*F[t], {2}]


and this:

expr2 = J'[t] == n*(y - L[t]) /. L[t] -> p0 + p1*J[t] - p2*F[t]


Transform your system to the one with two unknown functions J[t]and F[t].

It is already more simple. The outcome is, however, too long, so I do not write it here. You may evaluate the expression above. However, the structure of the system is as follows:

 DSolve[{J'[t] == a + b*J[t] + c*F[t],
F'[t] == d + f*J[t] + g*F[t]^2 + e*J[t]*F[t]}, {J, F}, t]


where I introduced new parameters ato einstead of yours, just to make the expression visible. Now I tried to solve it, but did not get the answer during first 5 minutes. It may sill come, of course, but more probable that Mma cannot solve it. And I would say, that it is most often useless to solve analytically equations with the huge number of parameters like in this case. Assume that you get the answer, it will be then tremendous, impossible to further analyze. To resume, I am pessimistic about the perspective to find this solution analytically.