Question

If f1(x,y) and f2(x,y) are homogeneous and of order ‘n’then the function f3(x,y) = f1(x,y) + f2(x,y) satisfies euler’s theorem.

(a) True

(b) False

I got this question in a national level competition.

This key question is from Euler’s Theorem in division Partial Differentiation of Engineering Mathematics

Answer 1

Correct choice is (a) True

The explanation is: Since f1(x,y) and f2(x,y) are homogeneous and of order n hence,

\(x \frac{∂f_1}{∂x}+y \frac{∂f_1}{∂y} = nf_1 (x,y)\)

\(x \frac{∂f_2}{∂x}+y \frac{∂f_2}{∂y} = nf_2 (x,y)\)

Hence adding these two equations,

We get

\(x \frac{∂f_1+f_2}{∂x}+y \frac{∂f_1+f_2}{∂y} = nf_2 (x,y)+nf_1 (x,y)\)

\(x \frac{∂f_3}{∂x}+y \frac{∂f_3}{∂y} = nf_3 (x,y)\)

Hence f3 satisfies euler’s theorem.