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Find the solution $y = f(x)$ of: $$ x^2 + y^2 - x^3 = 0 $$ Near the following points $(5,10)$,$(10,-30)$

I think I need to use the implicit function theorem and I tried this

  • First for the definition the function has continuous partial derivatives

  • Applying $(5,10)$: hence, $F(5,10) = 0$

  • $\frac{\partial}{\partial y}(5,10) = 2\cdot 10 = 20 \neq 0 $

Hence by the implicit function theorem it has a unique solution in the form of $y(x)$ around $(5,10)$

But what is this solution?

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  • $\begingroup$ Oh yes I forgot to add = 0 Thankss @lulu $\endgroup$ May 21 '19 at 20:03
  • $\begingroup$ What do you mean by solution? Do you want to express $y$ in terms of $x,$ or what? $\endgroup$
    – Allawonder
    May 21 '19 at 20:18
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The two solutions would be $y = \sqrt{x^3 - x^2}$ around $(5,10)$, and $y = -\sqrt{x^3 - x^2}$ around $(10, -30)$.

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If $x^2+ y^2- x^3= 0$ then $y^2= x^3- x^2$ so $y= \pm\sqrt{x^3- x^2}$. Since 10 is positive the branch that passes through (5, 10) is $y= \sqrt{x^3- x^2}$. Since -30 is negative the branch that passes through (10, -30) is $y= -\sqrt{x^3- x^2}$.

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Note that $$x^2 + y^2 - x^3 = 0 \implies y^2=x^3-x^2$$

Therefore you have two functions $$y=\pm \sqrt {x^3-x^2}$$

Therefore near the point $(5,10)$ you pick the positive sign and near the point $(10,-30)$ you pick negative sign.

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