I was teaching an ODE course last quarter and I came up with some fun exercises. I first asked the students to find the solutions for

\[\begin{aligned} \dot{x} &= 2xy \\ \dot{y} &= x^2+y^2 \end{aligned}\]

The trick here is consider new variables: \(z_1 = x+y, z_2=x-y\) The equations then become

\[\begin{aligned} \dot{z}_1 &= z_1^2 \\ \dot{z}_2 &= -z_2^2 \end{aligned}\]

which are easy to solve. Then I gave the students this ODE

\[\begin{aligned} \dot{x} &= 2xy \\ \dot{y} &= x^2+4y^2 \end{aligned}\]

If you proceed as in te first problem you are going to have a hard time. A better way is to consider the new variable \(z = y/x^2\) and we obtain that \(\dot{z} = 1\) and then with some creative substituting you get there.

If we would consider a more general ODE

\[\begin{aligned} \dot{x} &= \alpha xy \\ \dot{y} &= \beta x^2+ \gamma y^2 \end{aligned}\]

with \(\alpha, \beta, \gamma\) real parameters. Then it is easy to show that the first method works for \(\beta = \gamma\) , \(\alpha= 2 \beta\). The second method works for \(\gamma = 2 \alpha\).

So far I haven’t uncovered any other general case other then degenerate case obtained by setting a parameter to zero. I would love to know if this is part of some general theorem.