Can DSolve solve systems with unspecified function coefficients?

Question

I am an economic research student with no previous experience with Mathematica, so please pardon me if my questions sounds really stupid.

I am hoping to solve a system of nonlinear ODEs symbolically. I heard that the DSolve utility is very handy for this sort of problem, but before I invest my time in it, I hope to make sure that Mathematica is the answer I want.

Specifically, my system contains some function coefficients, but I do not know their explicit expression as of now, and they are just expressed as $\phi(t), \;\psi(t)$ etc. It also contains some unspecified constant coefficient as well. Can DSolve handle this situation?

If not, I'd very much appreciate it if someone could suggest any alternative methods.

Thank you!!

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Edit:

My system looks something like this:

$\beta\phi(t)\lambda(t)x(t)^{\alpha}y(t)^{\beta-1}-\psi(t)=0\\ \frac{dx}{dt}=\phi(t)x(t)^{\alpha}y(t)^{\beta}-\gamma{}x(t)\\ \frac{d\lambda}{dt}=k\psi(t)-\lambda(t)[\alpha\phi(t)x(t)^{\alpha-1}y(t)^{\beta}-\gamma]$

where $\lambda(t)\text{, }x(t)\text{ and }y(t)$ are the functions I want to solve for, and $\phi(t)$ and $\psi(t)$ are function coefficients without explicit expressions, $\alpha,\;\beta,\;\gamma$ and $k$ are unspecified coefficients.

I hope the solutions can be expressed in terms of integrals of $\phi(t)$ and $\psi(t)$. Is that possible?

Answer 1

It can solve some, can't solve others. Even if it can solve some sets of equations, make it a bit more complicated and it won't be able to anymore. So, I wouldn't bet on it being able to solve things analytically.

Here's an example where it can solve a nonlinear ODE:

DSolve[y'[t] == f[t]*y[t], y, t]

but, for example, it can't solve this

DSolve[
 {
  y'[t] == f[t] z[t],
  z'[t] == y[t]
  },
 {y, z},
 t
 ]

while it can solve, say, this

DSolve[
 {
  y'[t] == z[t],
  z'[t] == y[t]
  },
 {y, z},
 t
 ]

It can't give "closed-form" expressions for the system of equations you gave.