Question

$I$f $z=3x^3\sin \frac{y}{x}+(x^3+y^3)e^{y/x}$,

then $x^2\frac{{\partial }^{2}z}{\partial x^2}+y^2\frac{{\partial }^{2}z}{\partial y^2}+2xy\frac{{\partial }^{2}z}{\partial x\partial y}=?$

• A. $z$
• B. $2z$
• C. $4z$
• D. $6z$

Answer 1

The correct option is D $6z$

z is a homogeneous $e{q}^n$of degree$n=3$

$x^2\frac{d^{2}z}{d{x}^2}+2xy\frac{d^{2}z}{dxdy}+y^2\frac{d^{2}z}{d{y}^2}=n(n-1)z$

$x^2\frac{d^{2}z}{d{x}^2}+2xy\frac{d^{2}z}{dxdy}+y^2\frac{d^{2}z}{d{y}^2}=6z$