Система

[Гост]

[tex]\begin{array}{|l} \frac{1}{x}+\frac{1}{2y}=(x^{2}+3y^{2})(3x^{2}+y^{2}) \\ \frac{1}{x}-\frac{1}{2y}=2(y^{4}-x^{4}) \end{array}[/tex]

[Гост]

[tex]\begin{array}{|l} \frac{1}{x}+\frac{1}{2y}=(x^{2}+3y^{2})(3x^{2}+y^{2}) \\ \frac{1}{x}-\frac{1}{2y}=2(y^{4}-x^{4}) \end{array}[/tex]

Събираме и изваждаме 2-те у-я:

[tex]\begin{array}{|l} \frac{2}{x}=(x^{2}+3y^{2})(3x^{2}+y^{2})+2(y^{4}-x^{4}) \\ \frac{1}{y}=(x^{2}+3y^{2})(3x^{2}+y^{2})-2(y^{4}-x^{4}) \end{array}[/tex][tex]\Leftrightarrow[/tex][tex]\begin{array}{|l} \frac{2}{x}=x^{4}+10x^{2}y^{2}+5y^{4} \\ \frac{1}{y}=5x^{4}+10x^{2}y^{2}+y^{4} \end{array}[/tex]

[tex]x\ne0[/tex], [tex]y\ne0[/tex][tex]\Rightarrow[/tex][tex]\begin{array}{|l} 2=x^{5}+10x^{3}y^{2}+5xy^{4} \\ 1=5x^{4}y+10x^{2}y^{3}+y^{5} \end{array}[/tex]

Събираме и изваждаме:

[tex]\begin{array}{|l} 3=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5} \\ 1=x^{5}-5x^{4}y+10x^{3}y^{2}-10x^{2}y^{3}+5xy^{4}-y^{5} \end{array}[/tex][tex]\Leftrightarrow[/tex][tex]\begin{array}{|l} 3=(x+y)^{5} \\ 1=(x-y)^{5} \end{array}[/tex][tex]\Leftrightarrow[/tex][tex]\begin{array}{|l} \sqrt[5]{3}=x+y \\ 1=x-y \end{array}[/tex][tex]\Rightarrow[/tex][tex](x, y)=(\frac{\sqrt[5]{3}+1}{2}, \frac{\sqrt[5]{3}-1}{2})[/tex]