# Solve {(x^y=y^x),(x^2=y^3):} ?

Dec 10, 2016

((x=0, y=0), (x=1,y=1), (x=27/8, y = 9/4))

#### Explanation:

${x}^{y} = {y}^{x}$ admits as solution $x = y$

Now applying $\log$ to the equations

$\left\{\begin{matrix}{x}^{y} = {y}^{x} \\ {x}^{2} = {y}^{3}\end{matrix}\right.$-----(1)

we obtain

$\left\{\begin{matrix}y \log x = x \log y \\ 2 \log x = 3 \log y\end{matrix}\right.$

Dividing term to term we obtain the relationship

$\frac{y}{2} = \frac{x}{3}$

The solutions for (1) are obtained solving

$\left\{\begin{matrix}x = y \\ {x}^{2} = {y}^{3}\end{matrix}\right.$

and

$\left\{\begin{matrix}{x}^{2} = {y}^{3} \\ \frac{y}{2} = \frac{x}{3}\end{matrix}\right.$

so they are

((x=0, y=0), (x=1,y=1), (x=27/8, y = 9/4))

Attached a figure showing the interceptions

In red the two leafs of ${x}^{y} = {y}^{x}$
blue dotted $x = y$
blue continuous $\frac{y}{2} = \frac{x}{3}$ and
black ${x}^{2} = {y}^{3}$

The intersections have a black dot over.

The solution at $0 , 0$ can be understood as a regularization.