Consider the following differential system:
$$\begin{cases} x'=x-xy\\ y'= -y+xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$. How can I show that this system has a unique solution defined on a fixed interval $[0,T]$? I can't see how to apply Picard Lindelof theorem, because I can't prove that the function $f:[0,T]\times \mathbb{R}^2\to\mathbb{R}^2$, given by:
$$f(t,x,y)=(x-xy, -y+xy)$$ is uniform Lipschitz in $t$. Any ideas?
The system has a first integral $$ \frac{\dot y}{\dot x}=-\frac{y(x-1)}{x(y-1)}\implies \frac{\dot y}y(y-1)+\frac{\dot x}{x}(x-1)\implies y-\ln y+x-\ln x=C $$ and the level sets of that function are bounded. As solution curves follow the level sets, they are bounded too and thus solutions exist for all of $\Bbb R$.