# Elimination of Arbitrary Constants

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• Jan 31st 2011, 11:12 AM
sparky
Elimination of Arbitrary Constants
How do I eliminate the arbitrary constant in the equation y = ax^2 -1?

The answer at the back of the book is 2y = x(dy/dx)-2 however I don't understand how they arrived at this.

Here is my attempt:

y = ax^2 -1

dy/dx = 2ax

(dy/dx) (1/2x) = a

therefore y = (dy/dx) (1/2x)x^2 - 1

Where did I go wrong?
• Jan 31st 2011, 11:18 AM
Archie Meade
Quote:

Originally Posted by sparky
How do I eliminate the arbitrary constant in the equation y = ax^2 -1?

The answer at the back of the book is 2y = x(dy/dx)-2 however I don't understand how they arrived at this.

Here is my attempt:

y = ax^2 -1

dy/dx = 2ax

(dy/dx) (1/2x) = a

therefore y = (dy/dx) (1/2x)x^2 - 1

Where did I go wrong?

Stopping too soon ?

$\displaystyle \displaystyle\ y=\frac{dy}{dx}\left(\frac{1}{2x}\right)x^2-1$

$\displaystyle 2y=2\displaystyle\left[\frac{dy}{dx}\left(\frac{x}{2}\right)-1\right]$
• Jan 31st 2011, 11:23 AM
TheEmptySet
Quote:

Originally Posted by sparky
How do I eliminate the arbitrary constant in the equation y = ax^2 -1?

The answer at the back of the book is 2y = x(dy/dx)-2 however I don't understand how they arrived at this.

Here is my attempt:

y = ax^2 -1

dy/dx = 2ax

(dy/dx) (1/2x) = a

therefore y = (dy/dx) (1/2x)x^2 - 1

Where did I go wrong?

You didn't do anything wrong you just need to keep going. Your answer is equivalent.

$\displaystyle \displaystyle y=\frac{dy}{dx}\left( \frac{1}{2x}\right)x^2-1 \iff y=\left( \frac{x}{2} \right) \frac{dy}{dx}-1 \iff 2y=x\frac{dy}{dx}-2$

Edit: too slow
• Jan 31st 2011, 12:02 PM
sparky
Thanks for your replies TheEmptySet and Archie Meade.

Your workings have brought me closer to understanding this.

Why do we have to multiply both sides by 2?
• Jan 31st 2011, 12:25 PM
Archie Meade
Only to remove the "inconvenient" fraction (a half) on the right, nothing more!
Even without going that far, you were still correct where you finished up.
• Jan 31st 2011, 12:26 PM
sparky
Ok great, thanks a lot Archie Meade and TheEmptySet