Задача 72972 Решите систему уравнений: x3–y3=y5–x5,...

632f635587d19127bff140f0

01.10.2023 22:47:05

Решите систему уравнений: x3–y3=y5–x5, x2+y2=1

5f3ea7e3faf909182968ddd9

02.10.2023 11:55:01

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[m]\left\{\begin {matrix} x^3–y^3=y^5–x^5\\ x^2+y^2=1\end {matrix}\right.[/m]

[m]\left\{\begin {matrix} (x–y)(x^2+xy+y^2)=(y–x)(y^4+x^3y+x^2y^2+xy^3+y^4)\\ x^2+y^2=1\end {matrix}\right.[/m]

[m]\left\{\begin {matrix} (x–y)(1+xy)-(y–x)((x^4+y^4)+xy(x^2+y^2)+x^2y^2)=0\\ x^2+y^2=1\end {matrix}\right.[/m]

(x^2+y^2)^2=1^2 ⇒ x^4+2x^2y^2+y^4=1 ⇒ x^4+y^4=1-2x^2y^2

[m]\left\{\begin {matrix} (x–y)(1+xy)+(x–y)(1-x^2y^2+xy)=0\\ x^2+y^2=1\end {matrix}\right.[/m]

[m]\left\{\begin {matrix} (x–y)(1+xy+1-x^2y^2+xy)=0\\ x^2+y^2=1\end {matrix}\right.[/m]

[m]\left\{\begin {matrix} (x–y)(3-(x^2y^2-2xy+1)=0\\ x^2+y^2=1\end {matrix}\right.[/m]

[m]\left\{\begin {matrix} x–y=0\\ x^2+y^2=1\end {matrix}\right.[/m] или [m]\left\{\begin {matrix} 3-(x^2y^2-2xy+1)=0\\ x^2+y^2=1\end {matrix}\right.[/m]

[m]\left\{\begin {matrix} x=y\\ x^2+y^2=1\end {matrix}\right.[/m] или [m]\left\{\begin {matrix} (xy-1)^2=3\\ x^2+y^2=1\end {matrix}\right.[/m]