Question

verify euler's theorem from the function xy/x+y​

Answer 1

Euler's theorem

f(x,y)=x2+y21

f(tx,ty)=t2x2+t2y21=t1.f(x,y)=t−1f(x,y)

∴ f is a homogeneous function of degree −1 and by Euler's theorem

x∂x∂f+y∂y∂f=−f

Verification:

∂x∂f=2−1.(x2+y2)3/22x=(x2+y2)3/2−x

Similarly

∂y∂f=(x2+y2)3/2−y

x∂x∂f+y∂y∂f=−((x2+y2)3/2x2+y2)

x2+y2−1=−f