# How to solve this differential equation system?

I developed the following system of two ODEs while working on a problem of copulas:

f(u) (G(u) - G(0)) = 1,

g(v) (F(1) - F(v)) = 1

Here G is a primitive of g and F is a primitive of f.

I tried to solve the system via sage, which uses maxima for this, but maxima says it cannot solve the system. If that helps, one can assume that u and v belongs to the interval (0,1).

• How many variables do you have? – Igor Rivin Dec 7 '11 at 19:05

I'll rewrite this: let $x(t) = G(t) - G(0)$ and $y(t) = F(t) - F(1)$. Then the system says
$$y'(t) x(t) = 1,\ x'(t) y(t) = -1,\ x(0)=0,\ y(1) = 0$$
However, it's obviously impossible to satisfy the differential equations at $t=0$ or $t=1$. You say you want $u$ and $v$ to be in $(0,1)$, so maybe you could hope for $\lim_{t \to 0} x(t) = 0$ and $\lim_{t \to 1} y(t) = 0$. But that won't work either: the general solution of the system of differential equations is $x(t) = a e^{bt}$, $y = - \frac{e^{-bt}}{ab}$ for nonzero constants $a,b$, and these can't have limits of 0 at any finite $t$.