Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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).

share|improve this question
How many variables do you have? –  Igor Rivin Dec 7 '11 at 19:05
add comment

1 Answer

up vote 9 down vote accepted

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 $t$.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.