Coupled differential equations method

Question

$$\frac{dy}{dt} = (x-y)y$$

$$\frac{dx}{dt} = -y$$

How can I solve for $x(t), y(t)$? Is there a general method?

Answer 1

The first thing you might do is put it on the machine and see what it looks like. It seems that without initial conditions this is necessary, perhaps it's a well known system.

Answer 2

$$\frac{\mathrm{d}y}{\mathrm{d}x}=\displaystyle\frac{\frac{\mathrm{d}y}{\mathrm{d}t}}{\frac{\mathrm{d}x}{\mathrm{d}t}} = y-x.$$ This is now a simple first order problem, with solutions $y = c\mathrm{e}^x+x+1.$ I think @Alan got his equations mixed around.