Date: Sep 12, 2013 3:03 AM
Author: quasi
Subject: Re: Building an Equation to find (Maximum Y) ie Highest Point on a curve!
mervynmccrabbe@gmail.com wrote:

>

>x^4 + y^4 + a(x^2) - a(y^2) + 2(x^2)(y^2) - bxy + c = 0

>

>If I have correctly evaluated

>

>dy/dx = 4x^3 + 2ax + 4x(y^2) - by

No, that's not correct.

The correct result is:

dy/dx =

(4x^3 + 2ax + 4x(y^2) - by)/(-4y^3 + 2ay - 4y(x^2) + bx)

However, it _is_ true that dy/dx = 0 implies

4x^3 + 2ax + 4x(y^2) - by = 0

>If I could eliminate X and get a generalised Y-Only equation

>then I could manage the rest.

Ok, you asked for it ...

The system of equations

x^4 + y^4 + a(x^2) - a(y^2) + 2(x^2)(y^2) - bxy + c = 0

4x^3 + 2ax + 4x(y^2) - by = 0

yields the y-only equation

(d_8)(y^8)

+ (d_6)(y^6)

+ (d_4)(y^4)

+ (d_2)(y^2)

+ (d_0)

= 0

where

d_8 = 256b^2

+ 1024a^2

d_6 = -192(b^2)a

- 1024ac

- 768(a^3)

d_4 = -240(a^4)

- 168(a^2)(b^2)

+ 1920(a^2)c

- 27(b^4)

+ 288(b^2)c

+ 256(c^2)

d_2 = -16(a^5)

- 4(a^3)(b^2)

+ 384(a^3)c

- 1280a(c^2)

+ 144a(b^2)c

d_0 = 256(c^3)

+ 16(a^4)c

- 128(a^2)(c^2)

quasi