§ 44: 44.73

44.73 Розв’яжіть рівняння:

(x + y)² + 10(x + y)cos(πxy) + 25 = 0;

D = 100cos²(πxy) – 100 = 100(cos²(πxy) – 1) = –100sin²(πxy) ≤ 0;

при D < 0, ∅; при D = 0, 1 корінь;

–100 sin²(πxy) = 0;

1- cos(2πxy)2⁡ = 0;

cos(2πxy) = 1; 2πxy = 2πn, n ∈ Z; xy = n.

x + y = -10 cos(πxy)2 = –5 cos(πxy)

Якщо xy = n, –5cos(πn) = –5 • 1, cosπn = 1; –5 • 1, cosπn = –1.

x + y = 5; x = 5 – y; x = -5-25-4n2;

xy = n; y(5 – y) = n; y = -5+25-4n2;

x + y = –5; x = –5 – y; x = -5+25-4n2;

xy = n; y(–5 – y) = n; y = -5-25-4n2;

x = 5+25-4n2;

y = 5-25-4n2;

x = 5-25-4n2;

y = 5+25-4n2.