Question

Помогите!!! Решите систему уравнений:

$left { {{xy+ frac{y^3}{x}= frac{10}{3} } atop {xy+ frac{x^3}{y}=5 }} right.$

Answer 1

$left { {{xy+ frac{y^3}{x}= frac{10}{3} } atop {xy+ frac{x^3}{y}=5 }} right. \ left { {{ frac{x^2y+y^3}{x}= frac{10}{3} } atop { frac{xy^2+x^3}{y}=5 }} right. \ left { {{ frac{y(x^2y+y^2)}{x}= frac{10}{3} } atop { frac{x(y^2+x^2)}{y}=5 }} right. \ left { {{x^2+y^2=a geq 0} atop { frac{x}{y} =b}} right.$

$left { {{frac{a}{b}= frac{10}{3} } atop {ab=5 }} right. \ b=0.3a \ 0.3a^2=5 \ a= sqrt{ frac{5}{0.3} } = sqrt{ frac{50}{3} } \ b=0.3 sqrt{ frac{50}{3} }$

$left { {{x^2+y^2= sqrt{ frac{50}{3} } atop { frac{x}{y} =0.3 sqrt{ frac{50}{3} }} right. \ frac{x^2}{y^2} =frac{50cdot0.3^2}{3}=1.5 \ x^2=1.5y^2 \ 1.5y^2+y^2= sqrt{ frac{50}{3}} \ 2.5y^2= sqrt{ frac{50}{3}} \ y^2= frac{ sqrt{50} }{2.5 sqrt{3} }= sqrt{ frac{8}{3} } \ y_1= sqrt[4]{ frac{8}{3}} \ y_2= -sqrt[4]{ frac{8}{3}}$

$x^2=1.5 sqrt{ frac{8}{3} } = sqrt{ frac{18}{3} } = sqrt{6} \ x_1= sqrt[4]{6} \ x_2=- sqrt[4]{6}$

Ответ: $( sqrt[4]{6}; sqrt[4]{ frac{8}{3}})$; $( -sqrt[4]{6}; -sqrt[4]{ frac{8}{3}})$