Question

If u = ln (x^2 + y^2)/(x + y), find the constant k such that x dy/dx + y du/dy = k

Answer 1

well, just plug and chug:

u = ln(x^2+y^2) - ln(x+y)
∂u/∂x = 2x/(x^2+y^2) - 1/(x+y)
∂u/∂y = 2y/(x^2+y^2) - 1/(x+y)

x ∂u/∂x + y ∂u/∂y = 2x^2/(x^2+y^2) - x/(x+y) + 2y^2/(x^2+y^2) - y/(x+y)

= 2(x^2+y^2)/(x^2+y^2) - (x+y)/(x+y)
= 3

so, k=3