Solving a certain system of differential equations

Question

Let us consider the system of differential equations:
$$x'(t)= y(t)^2, y'(t) = x(t)^2, x(0) = a, y(0)=b, a, b \in \mathbb{R}.$$ How would one go about solving this system of differential equations?

Answer 1

Once you have $x^3 - y^3 = a^3 - b^3$, you solve that for one of the variables, say $y = (x^3 - a^3 + b^3)^{1/3}$. For convenience I'll write this as $y = (x^3 - c)^{1/3}$. Now your equation for $x$ says $$ x'(t) = (x^3-c)^{2/3}$$ This is separable, but unfortunately the integral necessary for the solution can't be done in closed form:

$$ t = \int \dfrac{dx}{(x^3-c)^{2/3}} + const$$