Z = y^2 - x^2 + y*ln(x/y)
dz/dx = -2x + y / (x/y)*1/y = -2x + y/x
dz/dy = 2y + ln(x/y) + y / (x/y)*(-x/y^2) = 2y + ln(x/y) - y/x*x/y =
= 2y + ln(x/y) - 1
Подставляем
x*dz/dx + y*dz/dy = -2x^2 + y + 2y^2 + y*ln(x/y) - y =
= -2x^2 + 2y^2 + y*ln(x/y) = y^2 - x^2 + y*ln(x/y) + y^2 - x^2 =
= z - x^2 + y^2
Где z = y^2- x^2 + y*ln(x/y)