Re: [Phys-L] solve equation w/o calculus

[Carl Mungan <mungan@usna.edu>]

> Note that if equation (4) had taken a slightly different form, such as
>
> > 	3x^2 - 2x√y + (4 - y^2) = 0			(4')
>
> then the place where the discriminant goes to zero would not be the
> extremum.

Let’s try both ways. Assume that the solution is restricted to occur at positive real values of x and y. (That is true for the actual physical problem from which the equation originated. Granted, I didn’t specify this restriction originally, but I hope you’ll allow it now.)

METHOD #1 - Rearrange (4’) into x = (Sqrt[y]+/-Sqrt[3y^2+y-12])/3. Discriminant goes to zero when y = (Sqrt[17]-1)/6 => x = Sqrt[(Sqrt[17]-1)/54].

METHOD #2 - Differentiate (4’) wrt x and put dy/dx=0 to get x = Sqrt[y]/3. Put that back into (4’) to get 3y^2+y-12 = 0.

Looks like both methods give the same solution to me…. -Carl

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Carl E. Mungan, Professor of Physics  410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363 
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