Question

Form the partial differential equation by eliminating the arbitrary function f from z=f(x²-y²)

Form the partial differential equation by eliminating the arbitrary function f from z=f(xy)

Form the partial differential equation by eliminating the arbitrary function from z²-xy=f(x/z)

Answer 1

1)

$Z=f(x^2-y^2)\\ \dfrac{\delta{Z}}{\delta{x}}=2xf'(x^2-y^2)\\ \dfrac{\delta{Z}}{\delta{y}}=-2yf'(x^2-y^2)\\ \therefore\\ y\dfrac{\delta{Z}} {\delta{x}}+x\dfrac{\delta{Z}}{\delta{y}}=0$

2)

$z=f(xy)\\ \dfrac{\delta{Z}}{\delta{x}}=yf'(xy)\\ \dfrac{\delta{Z}}{\delta{y}}=xf'(xy)\\ \therefore\\ x\dfrac{\delta{Z}}{\delta{x}}-y\dfrac{\delta{Z}}{\delta{y}}=0$

3)

$z^2-xy=f(\frac{x}{z})\\ 2z\dfrac{\delta{z}}{\delta{x}}-y=(\dfrac{1}{z}-\dfrac{x}{z^2}\dfrac{\delta{z}}{\delta{x}} )f'(\frac{x}{z})\\ 2z\dfrac{\delta{z}}{\delta{y}}-x=(-\dfrac{x}{z^2}\dfrac{\delta{z}}{\delta{y}} )f'(\frac{x}{z})\\ \therefore\\ \dfrac{\delta{z}}{\delta{y}}(\dfrac{xy}{z^2}-1)-\dfrac{x^2}{z^2}\dfrac{\delta{z}}{\delta{x}}=-\dfrac{x}{z}$