Correct Answer - Option 2 : 1
Concept:
A function f(x, y) is said to be homogeneous of degree n in x and y, it can be written in the form
f(λx, λy) = λn f(x, y)
Euler’s theorem:
If f(x, y) is a homogeneous function of degree n in x and y and has continuous first and second-order partial derivatives, then
\(x\frac{{\partial f}}{{\partial x}} + y\frac{{\partial f}}{{\partial y}} = nf\)
\({x^2}\frac{{{\partial ^2}f}}{{\partial {x^2}}} + 2xy\frac{{{\partial ^2}f}}{{\partial x\partial y}} + {y^2}\frac{{\partial f}}{{\partial y}} = n\left( {n - 1} \right)f\)
If z is homogeneous function of x & y of degree n and z = f(u), then
\(x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}} = n\frac{{f\left( u \right)}}{{f'\left( u \right)}}\)
Calculation:
Given, \(u = log\left( {\frac{{{x^2} + {y^2}}}{{x + y}}} \right)\)
\(z = \frac{{{x^2} + {y^2}}}{{x + y}}\)
z is a homogenous function of x & y with a degree 1.
Now, z = eu
Thus, by Euler’s theorem:
\(x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}} = \frac{{{e^u}}}{{{e^u}}}\)
\(x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}} = 1\)
