Question

If $f(x,y)=x\cos \left(\frac{y}{x}\right)+y\tan \left(\frac{y}{x}\right)$, then $x^2{f}_{xx}+2xy{f}_{xy}+y^2{f}_{yy}=?$

• A. $0$
• B. $1$
• C. $f(x,y)$
• D. $2f(x,y)$

Answer 1

The correct option is A $0$

We know that if $u=x\varphi \left(\frac{y}{x}\right)-y\psi \left(\frac{y}{x}\right)$, then $x^2{u}_{xx}+2xy{u}_{xy}+y^2{u}_{yy}=0$

here, $\varphi \left(x\right)=\cos x$ and $\psi \left(x\right)=\tan x$

Therefore, the answer is $0$.