Solve in positive integers the equation $a^3+b^3=9ab$ Solve in positive integers the equation:
$$a^3+b^3=9ab$$
I try to:
$$\dfrac{a^2}{b}+\dfrac{b^2}{a}=9\Longrightarrow a^2<9b,b^2<9a$$
Of course, I can't solve it. Can anyone help?

HINT: $x^3+y^3=9xy$ is the Cartesian equation of a Folium of Descartes $x^3+y^3=3axy$ with $a=3$. This is a rational cubic with a double point and its parametric equations are $x=\frac{9t}{1+t^3}$ and $y=tx$ The loop of the curve occurs for the values $t\gt 0$ so the integer points we are looking for belong to this loop.

For $t\ge 3$ one has $x\lt 1$ hence possible integers values $x, y$ correspond to values of the parameter $t$ such that $0\lt t \lt 3$.
When $t=1$ there is not integer solution $(a,b)$ but when $t=2$ there is the solution $\color {red}{(a,b)=(2,4)}$ (and trivially $(b,a)$ from symmetry over $y=x$).
Now,to discard fractional values of $t$, taking derivatives, we have
$$x’=\frac{9-18t^3}{(1+t^3)^2}$$
$$y’=\frac{18t-9t^4}{(1+t^3)^2}$$ it follows
$$\max x=\frac{9\sqrt[3]{2}}{1+2}=3\sqrt[3]{2}\approx 3.7797$$
$$\max y=\frac{9\sqrt[3]{4}}{1+2}=3\sqrt[3]{4}\approx 4.7622$$
Consequently there are just $12$ points to verify (in the “lattice” $\{1,2,3\}\text {x}\space \{1,2,3,4\}$). Thus the only solution in positive integers is the given one $(2,4)$.
