Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am quite new to differential equations and derivatives. I want to derive an differential form for equation of an ellipse. If i start with an ordinary ellipse equation

\begin{equation} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \end{equation}

How do i derive it then to get this form

$$ -\frac{dx}{dy} = \frac{a^2}{b^2} \frac{y}{x} $$

I would need an equation and some brief explanation on the procedure.

share|cite|improve this question
An ellipse does not hace a differential equation... What exactly are you trying to do? – Mariano Suárez-Alvarez Apr 5 '12 at 22:56
I think perhaps "derive" means take derivatives? – Alex R. Apr 5 '12 at 23:01
Have you encountered "implicit differentiation?" Here is a helpful link: – Alex R. Apr 5 '12 at 23:03
up vote 6 down vote accepted

The equation $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \tag{1}$$ has two variables: $\{ x, y \}.$ By "derive," it seems that you mean $ \frac{dx}{dy}.$

Well, differentiating equation $(1)$ w.r.t $y,$ we get: $$ \frac{d}{dy} \frac{x^2}{a^2} + \frac{d}{dy} \frac{y^2}{b^2} = \frac{d}{dy} 1 \tag{2}$$

First, note that $\dfrac{d}{dy} 1 = 0,$ $\dfrac{d}{dy} y = 1,$ and $\dfrac{d}{dy} f^2 = 2 f \dfrac{df}{dy}.$


  1. $ \dfrac{d}{dy} \dfrac{x^2}{a^2} = 2 \dfrac{x^{2-1}}{a^2} \dfrac{dx}{dy}$

  2. $ \dfrac{d}{dy} \dfrac{y^2}{b^2} = 2 \dfrac{y^{2-1}}{b^2} \dfrac{dy}{dy} $

In other words, equation $(2)$ becomes:

$$ 2\frac{x}{a^2} \frac{dx}{dy} + 2 \frac{y}{b^2} = 0. $$

The rest is simple algebra, you can isolate $\dfrac{dx}{dy}$ one side, and get: $$ -\frac{dx}{dy} = \frac{a^2}{b^2} \frac{y}{x} $$

share|cite|improve this answer
Oh. I forget to add, $a, b$ are constants, and the differential operator is linear. – user2468 Apr 5 '12 at 23:40
Thank you, this is a good explanation. – 71GA Apr 10 '12 at 7:17

The answer you want is actually not the differential equation of the family of ellipse. A differential equation is free of arbitrary constants like $a$ and $b$. Since there are two arbitrary constants, you need to differentiate 2 times (the order of the differential equation should be 2). The answer you want is just the negative of slope of normal at any point $(x,y)$ on the ellipse.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.