Solution to paired exponential growth

Question

I have a pair of equations such that:

$$ \frac{dx}{dt} = x ( r_x + \alpha y) \\ \frac{dy}{dt} = y ( r_y + \beta x) $$

where $r_x , \alpha, r_y \ \text{and} \ \beta$ are constants. I want to find the general solution (i.e. $x(t)$ and $y(t)$) of these equations. So far been I unable to solve them and I'm not sure how to get a general solution or if such a solution exists.

Answer 1

Too long for a comment, take it as a hint.

Starting from:

$$ x' = x ( r_x + \alpha y) \\ y' = x ( r_y + \beta y) $$

Differentiate the first:

$$ x'' = x' ( r_x + \alpha y) + x \alpha y' $$ and eliminate $y'$

$$ x'' = x' ( r_x + \alpha y) + \alpha x^2 ( r_y + \beta y) $$

Now solve $y$ from the first:

$$ y=\frac{1}{ \alpha}\frac {x'}{ x}- r_x $$

So

$$ x'' = x' ( r_x (1-\frac{1}{ \alpha})+ \frac {x'}{ x} ) + \alpha x^2 ( r_y +\frac{ \beta}{ \alpha}\frac {x'}{ x}- \beta r_x ) $$

Now, you can solve it... maybe.

Answer 2

EDITED: With the new second equation, $$ x = \dfrac{y'}{\beta y} - \dfrac{r_y}{\beta} $$ and then $$y'' = - \alpha r_y y^2 + (\alpha y' - r_x r_y) y + r_x y' + (y')^2/y $$ and $$ t + c_1 = \int \dfrac{dy}{y r_y \left(W\left(c_2 y^{r_x/r_y} \exp( \alpha y/r_y)\right)+1\right)} $$ where $W$ is the Lambert W function.