system of non-linear differential equations $x'=y^2$ and $y'=x^2$

Question

I am working on Exercise 2.90 in Jeffrey M. Lee's book Manifolds and Differential Geometry. To solve it, I need to solve the following system of non-linear differential equations: $$ \begin{array}{rcl} \displaystyle \frac{dx}{dt}&=&y^{2} \\ \displaystyle \frac{dy}{dt}&=&x^{2} \end{array} $$ Integrating the second equation and substitution into the first equation gives me $$ dx=\left(\int{}x^{2}\,dt\right)^{2}dt $$ but I don't know where to go from there.

Answer 1

One approach is to guess at the form of solutions. I tried exponentials and products and that didn't seem to work. Polynomials went nowhere. But then I tried power functions. A solution is $$ x = \frac{-1}{t}, y = \frac{-1}{t}$$

Answer 2

Rather than guess, you can also look at $dy/dx = (dy/dt)/(dx/dt)$ which leads to $$ \frac{dy}{dx} = \frac{x^2}{y^2} \implies y^2 dy = x^2 dx \implies y^3 = x^3 + C. $$