How to Form Differential Equations of Ellipse?

Ellipse can be defined as a curve which is comprised of a cone and Intersection of plane or we can say it is kind of conic section. Circular cone and plane intersect in such a way that they create a closed curve. This closed curve is called as ellipse. When axis of circular cone is orthogonal to plane which is to be intersected, ellipse becomes a Circle that is the circle is created from ellipse in a special condition. An ellipse has two radius of different magnitude. Let’s consider two radii are ‘a’ and ‘b’, then ellipse will be represented graphically as shown below.
 
 
 
Basic equation of an ellipse is given by
x²/ a² + y² / b² = 1,
Let’s try to understand how to form Differential Equations of ellipse? That is we have to find out the value of d x / d y.
For this rewrite the ordinary equation for ellipse as:
x2/ a2 + y2 / b2 = 1, in above equation there are two variables (x and y). Lets differentiate the above equation with respect to 'y', we get:
d [(x²/ a² + y² / b²)] / dx = d (1) / d y.
 
Differentiation of a constant is always zero hence d (1) / d y = 0,
Above equation can be written as:
d (x²/ a²) / d y + d (y² / b²) / d y = 0,
Differentiation rule says that d (x n) / d y = n x n – 1 (d x / d y).
Applying this rule in above equation, we get:
2 x / a ² (d x / d y) + 2 y / b ² = 0            (since d y / d y = 1),
Taking LCM as 2b 2 gives 2 x b ² (d x / d y) + 2 y a ² = 0,
On rearranging above equation, we get:
2 x b ² (d x / d y) = - 2 y a ²,
Then solving for (d x / d y), we get
d x / d y = - (a ² / b ²) (y / x).