How to solve the following equations on $\mathbb{R}$?(the result must be expressed by elementary function)

Question

Solve the following equations on $\mathbb{R}$(the result must be expressed by elementary function) \begin{eqnarray} \begin{cases} x^5-5x^3y+5x^2+5xy^2-5y+1=0 \\ y^5-5xy^3+5y^2+5x^2y-5x+2=0 \ \end{cases} \end{eqnarray}

(I have input it into wolfram alpha only get the approximate solution.This problem is provided by a chinese tieba(one kind of BBS) user "我不知道我不知道”）

(tips: let $x=a+b+c,y=1/a+1/b+1/c$ and $abc=1$)

Answer 1

After a hard search, I contacted the author of the problem. He did it in this way

Let $x=a+b+c,y=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ and $abc=1$

$(a+b+c)^5-5(a+b+c)^3(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})+5(a+b+c)^2+5(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^2-5(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})+1=0 :A$

$(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^5-5(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^3+5(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^2+5(a+b+c)^2(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})-5(a+b+c)+2=0 :B$

Through complex operations on $A$ and $B$ and $abc=1$,we find $a^5,b^5,c^5$ is the three different roots of equation $t^3+t^2-2t-1=0$

Then let $t=m+\frac{1}{m}$,we get $m^7=1$ and $m\neq1$.

let $ω_1=e^\frac{2πi}{7}$,$ω_2=e^\frac{4πi}{7}$,$ω_3=e^\frac{6πi}{7}$

then $a=(ω_1+\frac{1}{ω_1})^{0.2}$ ,$b=(ω_2+\frac{1}{ω_2})^{0.2}$ ,$c=(ω_3+\frac{1}{ω_3})^{0.2}$

and $x=a+b+c,y=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Answer 2

In $\mathbb{R}$, there is only one solution.

From the first equation, we have $$y=\frac{5 x^3+5+\sqrt{5} \sqrt{x^6-10 x^3-4 x+5}}{10 x}$$ Plugging in the second, we need to solve for $x$ $$x^{25}-50 x^{22}+5 x^{20}+875 x^{19}-75 x^{17}-6750 x^{16}+1260 x^{15}+125 x^{14}+23125 x^{13}-$$ $$17425 x^{12}+1625 x^{11}-31240 x^{10}+63250 x^9+11875 x^8+24550 x^7-$$ $$29625 x^6-16245 x^5+32750 x^4-11250 x^3+26325 x^2-22375 x+6876=0$$ which does not show any rational solution.

Its closest approximation is $$x=-\sin ^{\frac{24}{7}}(e \pi ) (-\cos (e \pi ))^{3/7} \sec ^6(e \pi )\,\pi ^{\frac{3(9 e-2)}{7} }\,\exp\Bigg[-\frac{10 e+8 \pi -9 e \pi -12 e^2 \pi +34 e \pi ^2}{7 e \pi } \Bigg]$$ which is in an error of $3.89\times 10^{-18}$

Plugging these $(x,y)$ in the equations, the rhs are $0$ (for sure) and $-6.456\times 10^{-17}$.