Get a free home demo of LearnNext

Available for CBSE, ICSE and State Board syllabus.
Call our LearnNext Expert on 1800 419 1234 (tollfree)
OR submit details below for a call back


Differential Equations of Family of Curves

Have a doubt? Clear it now.
live_help Have a doubt, Ask our Expert Ask Now
format_list_bulleted Take this Lesson Test Start Test

Differential Equations of Family of Curves - Lesson Summary

Differential equations are used to depict some geometrical properties of curves.


The equation of a circle whose centre is at the origin is x2 + y2 = r2

Differentiating with respect to x:

2x + 2y dy/dx = 0

⇒ x + y dy/dx = 0 [Required differential equation]

⇒ x = - y dy/dx ⇒ x dx = - y dy

Understanding how the differential equation represents family of circles:

⇒ ∫ x dx = - ∫ y dy

⇒ x2/2 = - y2/2 + C1

⇒ x2 = - y2 + 2C1

⇒ x2 + y2 = C, Where C =  2C1, C ∈ R


1. The equation representing a family of curves with one unknown parameter:

F1(x,y,a) = 0 ... (1)

2. Differentiating F1(x,y,a) : g(x,y,y',a) = 0 ......(2)

3. Eliminating a between (1) and (2):

F(x,y,y') = 0 ......(3) [Required differential equation]

Observe that the differential equation obtained does not contain unknown parameter a, and contains the derivative.

The equation representing a family of curves with two unknown parameters:

F2(x,y,a,b) = 0 .....(1). Here, 'a' and 'b' are the unknown parameters.

2. Differentiating F2(x,y,a,b): g(x,y,y',a,b) = 0 .....(2)

Since there are two unknown parameters, a and b, two equations, one and two, are not sufficient to eliminate them.

3. Differentiating g(x,y,y',a,b): h(x,y,y',y",a,b) = 0 ......(3)

3. Eliminating a and b between (1), (2) and (3):

F(x,y,a,b) = 0 [Required differential equation]

Note: The number of unknown parameters determines the order of the differential equation. If the number of unknown parameters present in the equation corresponding to the family of curves is two, then the order of the differential equation obtained is two.


Feel the LearnNext Experience on App

Download app, watch sample animated video lessons and get a free trial.

Desktop Download Now
Try LearnNext at home

Get a free home demo. Book an appointment now!