conservation law for the trajectories

Question

Is it possible to find the conservation equation as the form of $Q=h(x,y)$, given that

$$\dot{x}=x-xy$$ $$\dot{y}=5xy-5y$$

I am not sure how to start with.

Answer 1

"Divide" one equations by another and find a separable equation $$ \frac{dy}{dx}=\frac{5xy-5y}{x-xy},\, $$ integrate.

Answer 2

1. Construct from the given coupled ODEs $$\tag{1} \dot{x}~=~f(x,y)~:=~x(1-y), \qquad \dot{y}~=~g(x,y)~:=~5(x-1)y, $$ an inexact differential $$\tag{2} \omega~:=~g(x,y)\mathrm{d}x-f(x,y)\mathrm{d}y.$$

2. Find an integrating factor $\lambda$ to the inexact differential (2) such that $$\tag{3}\mathrm{d}(\lambda \omega)~=~0.$$

3. Poincare Lemma then implies that there locally exists a function $h$ such that $$\tag{4}\lambda \omega~=~\mathrm{d}h. $$

4. Argue why such $h$ from eq. (4) would be constant/conserved along a trajectory of the systems of coupled ODEs (1).

5. Show that $$\tag{5}\lambda ~=~\frac{1}{xy}, \qquad x~\neq~ 0~\neq~ y,$$ is an integrating factor.

6. Determine from eq. (4) that $$\tag{6}h(x,y)~=~5x+y-\ln|x^5y|, \qquad x~\neq~ 0~\neq~ y,$$ is a conserved quantity.

7. We can define a globally defined conserved quantity as $$\tag{7} \tilde{h}(x,y)~=~e^{-h(x,y)}~=~x^5ye^{-5x-y}.$$