\begin{equation} \dot{x}(t)=\left(1-\frac{x(t)}{E_{0}}\right) x(t) - a\ y(t) \end{equation} \begin{equation} \dot{y}(t)=\left(1-\frac{a\ y(t)}{x(t)}\right) y(t) \end{equation}

I am trying to solve the system of differential equations above, but (I think) it is nontrivial to calculate its solution. Any suggestions?

up vote 1 down vote accepted

Solve the first equation for $y$, substitute into the second. You get a second-order differential equation for $x(t)$:

$$ - \left( {\frac {d^{2}}{d{t}^{2}}}x \left( t \right) \right) x \left( t \right) {E_{{0}}}^{2}+ \left( {\frac {d}{dt}}x \left( t \right) \right) ^{2}{E_{{0}}}^{2}- \left( x \left( t \right) \right) ^{3}E_{{0}}+ \left( x \left( t \right) \right) ^{4} =0$$

Maple solves this explicitly: $$x \left( t \right) =4\,{\frac {{{\rm e}^{ta+b}}{a}^{2}{E_{{0}}}^{2}}{- 4\,{a}^{2}{E_{{0}}}^{2}+{{\rm e}^{2\,ta+2\,b}}+4\,{{\rm e}^{ta+b}}E_{{0 }}+4\,{E_{{0}}}^{2}}} $$ where $a$ and $b$ are arbitrary constants.

  • The solution $(x(t),y(t))=(E_0,0)$ corresponds to the limit $a=1$, $b\to-\infty$. – Did Jul 1 '12 at 20:22

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