How to solve this differential equation system? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T00:25:40Z http://mathoverflow.net/feeds/question/82897 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82897/how-to-solve-this-differential-equation-system How to solve this differential equation system? Kjetil B Halvorsen 2011-12-07T18:59:57Z 2011-12-07T22:13:52Z <p>I developed the following system of two ODEs while working on a problem of copulas:</p> <p>f(u) (G(u) - G(0)) = 1, </p> <p>g(v) (F(1) - F(v)) = 1</p> <p>Here G is a primitive of g and F is a primitive of f.</p> <p>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).</p> http://mathoverflow.net/questions/82897/how-to-solve-this-differential-equation-system/82900#82900 Answer by Robert Israel for How to solve this differential equation system? Robert Israel 2011-12-07T19:19:26Z 2011-12-07T19:19:26Z <p>I'll rewrite this: let $x(t) = G(t) - G(0)$ and $y(t) = F(t) - F(1)$. Then the system says</p> <p>$$y'(t) x(t) = 1,\ x'(t) y(t) = -1,\ x(0)=0,\ y(1) = 0$$</p> <p>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 $t$.</p>