if-x-3-y-3-3axy-find-dy-dx-in-terms-of-x-and-y-and-prove-that-dy-dx-cannot-be-equal-to-1-for-finite-values-of-x-and-y-except-x-y-please-help-

Question Number 13226 by chux last updated on 16/May/17

if x^{3} + y^{3} = 3 axy, find dy / dx in terms

of x and y and prove that dy / dx

cannot be equal to - 1 for finite

values of x and y except x = y .

please help

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/May/17

3 x^{2} + 3 y^{2} \frac{d y}{d x} = 3 a y + 3 a x \frac{d y}{d x} (\frac{d y}{d x} = y^{^{'}})

\Rightarrow (3 y^{2} - 3 a x) y^{^{'}} = 3 (a y - x^{2}) \Rightarrow

y^{^{'}} = \frac{a y - x^{2}}{y^{2} - a x} .

i f : (y^{^{'}} = - 1) \Rightarrow a y - x^{2} = a x - y^{2} \Rightarrow

(y - x) (y + x) + a (y - x) = 0 \Rightarrow (y - x) (y + x + a) = 0 \Rightarrow

\Rightarrow {\begin{cases} y - x = 0 \Rightarrow y = x \\ y + x + a = 0 \Rightarrow y = - (x + a) . \end{cases}

Answered by ajfour last updated on 16/May/17

x^{3} + y^{3} = 3 a x y \dots \dots (i)

3 x^{2} + 3 y^{2} \frac{d y}{d x} = 3 a y + 3 a x \frac{d y}{d x}

\frac{d y}{d x} = \frac{a y - x^{2}}{y^{2} - a x}

\frac{d y}{d x} + 1 = \frac{a y - x^{2} + y^{2} - a x}{y^{2} - a x}

= \frac{(y - x) (x + y + a)}{y^{2} - a x}

\Rightarrow \frac{d y}{d x} + 1 = 0 o n l y w h e n

x = y, o r (x + y + a) = 0

{(x + y)}^{3} = x^{3} + y^{3} + 3 x y (x + y) \dots (i i)

x^{3} + y^{3} = 3 a x y \dots (i)

(i) + (i i) :

{(x + y)}^{3} = 3 x y (x + y + a)

\Rightarrow i f (x + y + a) = 0, x + y = 0

s o f o r (x + y + a) = 0 c o n d i t i o n i s

x + y = 0 a n d e v e n a = 0

t h e r e f o r e i f a \neq 0, x + y + a \neq 0

\Rightarrow \frac{d y}{d x} + 1 = 0 o r \frac{d y}{d x} = - 1 o n l y

f o r x = y . (g r a n t e d t h e y a r e f i n i t e)

Commented by chux last updated on 17/May/17

thanks alot .

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