If-z-z-2-3x-3y-0-prove-that-2-z-x-2-2-z-y-2-2z-x-1-z-2-x-3-please-help-

Question Number 89178 by necxxx last updated on 15/Apr/20

I f z (z^{2} + 3 x) + 3 y = 0 p r o v e t h a t

\frac{\partial^{2} z}{\partial x^{2}} + \frac{\partial^{2} z}{\partial y^{2}} = \frac{2 z (x - 1)}{{(z^{2} + x)}^{3}}

p l e a s e h e l p .

Commented by niroj last updated on 16/Apr/20

If z (z^{2} + 3 x) + 3 y = 0 P rove that

\frac{\partial^{2} z}{\partial x^{2}} + \frac{\partial^{2} z}{\partial y^{2}} = \frac{2 z (x - 1)}{{(z^{2} + x)}^{3}}

D . w . r . to x

z^{3} + 3 xz + 3 y = 0

{3 z}^{2} . \frac{dz}{dx} + 3 z + 3 x \frac{dz}{dx} = 0

3 \frac{dz}{dx} (z^{2} + x) = - 3 z

\frac{dz}{dx} = \frac{- z}{z^{2} + x} =

\frac{d^{2} z}{{dx}^{2}} = \frac{- \frac{dz}{dx} (z^{2} + x) + z (2 z . \frac{dz}{dx} + 1)}{{(z^{2} + x)}^{2}}

= \frac{- \frac{dz}{dx} (z^{2} + x) + {2 z}^{2} \frac{dz}{dx} + z}{{(z^{2} + x)}^{2}}

= \frac{\frac{dz}{dx} ({2 z}^{2} - z^{2} - x) + z}{{(z^{2} + x)}^{2}} = \frac{\frac{- z}{z^{2} + x} (z^{2} - x) + z}{{(z^{2} + x)}^{2}}

= \frac{xz - z^{3} + z^{3} + zx}{{(z^{2} + x)}^{3}} = \frac{2 zx}{{(z^{2} + x)}^{3}}

Again,

D . w . r . to . y

z^{3} + 3 xz + 3 y = 0

{3 z}^{2} \frac{dz}{dy} + 3 x \frac{dz}{dy} + 3 = 0

3 \frac{dz}{dy} (z^{2} + x) = - 3

\frac{dz}{dy} = - 1 . {(z^{2} + x)}^{- 1}

\frac{d^{2} z}{{dy}^{2}} = - 1 . (- 1) {(z^{2} + x)}^{- 2} . 2 z \frac{dz}{dy}

= \frac{2 z}{{(z^{2} + x)}^{2}} . \frac{- 1}{(z^{2} + x)}

= \frac{- 2 z}{{(z^{2} + x)}^{3}}

now,

\frac{\partial^{2} z}{\partial x^{2}} + \frac{\partial^{2} z}{\partial^{2} y} = \frac{2 xz}{{(z^{2} + x)}^{3}} + \frac{- 2 z}{{(z^{2} + x)}^{3}}

= \frac{2 xz - 2 z}{{(z^{2} + x)}^{3}} = \frac{2 z (x - 1)}{{(z^{2} + x)}^{3}}

hence

∴ \frac{\partial^{2} z}{\partial x^{2}} + \frac{\partial^{2} z}{\partial y^{2}} = \frac{2 z (x - 1)}{{(z^{2} + x)}^{3}} Proved / / .

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