Formation of Differential Equation

Here you will learn formation of differential equation with examples.

Let’s begin –

Formation of Differential Equation

Algorithm

1). Write the equation involving independent variable x (say), dependent variable y (say) and the arbitrary constants.

2). Obtain the number of arbitrary constants in Step 1. Let there be n arbitrary constants.

3). Differentiate the relation in step 1 n times with respect to x.

4). Eliminate arbitrary constants with the help of n equations involving differential coefficients obtained in step 3 and an equation in step 1. The equation so obtained is the desired differential equation.

Example : Form the differential equation of the family of curves represented by c(y+c)2 = x3 , where c is a parameter.

Solution : The equation of the family of curves is c(y+c)2 = x3                            ……….(i)

Clearly, it is one parameter family of curves, so we shall get a differential equation of first order.

Differentiating (i) with respect to x, we get

2c(y + c) dydx = 3x2                          ………(ii)

Dividing (i) by (ii), we get

c(c+y)22c(y+c)dydx = x33x2

y + c = 2x3dydx

c = 2x3dydx – y

Substituting the value of c in (i), we get

(2x3dydx – y)(2x3dydx)2 = x3

827x(dydx)349(dydx)2y = x

8x(dydx)312y(dydx)2 = 27x

This is the required differential equation of the curves represented by (i).

Example : Form the differential equation representing the family of curves y = A cos(x + B), where A and B are parameter.

Solution : The equation of the family of curves is y = A cos(x + B)                            ……….(i)

This equation contains two arbitratry constants. So, let us differential it two times to obtain a differential equation of second order.

Differentiating (i) with respect to x, we get

dydx = -A sin(x + B)                          ………(ii)

Differentiating (ii) with respect to x, we get

d2ydx2 = -A cos(x + B)

  d2ydx2 = -y             [Using (i)]

  d2ydx2 + y  = 0, which is the required differential equation of the given family of curves.                

Leave a Comment

Your email address will not be published. Required fields are marked *