Question

If $u={\tan }^{-1}\left(\frac{y}{x}\right)$, then by Euler’s theorem the value of $x\frac{\partial u}{\partial y}+y\frac{\partial u}{\partial y}=$

• A. $\sin \left(u\right)$
• B. $\tan \left(u\right)$
• C. $0$
• D. $\cos \left(u\right)$

Answer 1

The correct option is C

$0$

Explanation for the correct option:

Finding the value of the expression:

Given equation,

$u={\tan }^{-1}\left(\frac{y}{x}\right)$

$=x^{0}f\left(\frac{y}{x}\right)$

From the above equation, it is clear that $u$ is a homogeneous function of the form $x^{n}f\left(\frac{y}{x}\right)$.

Here, $n=0$

Using Euler’s theorem, we get

$x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=nu$

$=0$

Hence, the correct option is C.