# Can DSolve solve systems with unspecified function coefficients?

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!!

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?

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This is very vague. Can you at least provide your equations? –  Vitaliy Kaurov Aug 3 '12 at 7:31
@VitaliyKaurov Thanks for the advice! –  Vokram Aug 3 '12 at 7:49
Here you can find which types of equations Mathematica can solve. If I'm not mistaken your equations are to be found in the first-order ODE or more probably the systems of ODEs section (subsection Linear system of ODEs). However, I have never seen examples like yours with unspecified functions, so I don't think that MMA can handle that. I would be pleasantly surprized learning that it can, though. –  Sjoerd C. de Vries Aug 3 '12 at 11:21
I hope the solutions can be expressed in terms of integrals of ϕ(t) and ψ(t). Is that possible? I think that for arbitrary $\alpha$, $\beta$ it is highly improbable. –  Andrew May 17 '13 at 8:06
From the other hand if to introduce a new function $z(t)=x(t)^\alpha y(t)^\beta$ or $z(t)=x(t)^{\alpha-1} y(t)^\beta$ then it is possible to express $x,y,\lambda$ as some integrals of $\phi$, $\psi$ and $z$. –  Andrew May 17 '13 at 8:21

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.

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