Equation in rational numbers

Question

I can't seem to find the way to solve the following equation so help would be much appreciated..

$x^2+y^2=x^3+y^3$ over $\mathbb{Q}$

Answer 1

Take $\lambda=(t^2+s^2)/(t^3+s^3)$ for some $ t,s \in \Bbb Z$ such that $s \neq -t$ then $(\lambda t,\lambda s)$ is a solution.If $(x,y)$ is a non-trivial solution take $\lambda'=y/x$ ( Note that $\lambda' \neq 0$ )then $$x^3+y^3=x^2+y^2$$ $$\implies x(\lambda'^3+1)=(\lambda'^2+1)$$ $$\implies x=(\lambda'^2+1)/(\lambda'^3+1)$$ Put $\lambda'=s'/t'$ for $s',t' \in \Bbb Z$. We get the same form as above so those are the only solutions.