Question #228418

Form the partial differential equation by eliminating the arbitrary function f from z=f(x^{2}-y^{2})

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 z^{2}-xy=f(x/z)

Expert's answer

1)

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

2)

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

3)

"z^2-xy=f(\\frac{x}{z})\\\\\n2z\\dfrac{\\delta{z}}{\\delta{x}}-y=(\\dfrac{1}{z}-\\dfrac{x}{z^2}\\dfrac{\\delta{z}}{\\delta{x}} )f'(\\frac{x}{z})\\\\\n2z\\dfrac{\\delta{z}}{\\delta{y}}-x=(-\\dfrac{x}{z^2}\\dfrac{\\delta{z}}{\\delta{y}} )f'(\\frac{x}{z})\\\\\n\\therefore\\\\\n\\dfrac{\\delta{z}}{\\delta{y}}(\\dfrac{xy}{z^2}-1)-\\dfrac{x^2}{z^2}\\dfrac{\\delta{z}}{\\delta{x}}=-\\dfrac{x}{z}"

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