The order of the differential equation to be formed is equal to the number of arbitrary constants present in the equation of the family of curves.

**Formation of ordinary differential equation:**

Consider the equation *f* (
*x*, *y* ,*c*_{1} )
= 0 -------(1) where *c*_{1}
is the arbitrary constant. We form the differential equation from this
equation. For this, differentiate equation (1) with respect to the independent
variable occur in the equation.

Eliminate the arbitrary constant *c* from (1) and its
derivative. Then we get the required differential equation.

Suppose we have *f* (
*x*, *y* ,*c*_{1} ,*c*_{2} ) = 0 . Here we have two
arbitrary constants *c*_{1} and *c*_{2} . So, find
the first two successive derivatives. Eliminate *c*_{1} and *c*_{2}
from the given function and the successive derivatives. We get the required
differential equation.

**Note**

The order of the differential equation to be formed is equal to
the number of arbitrary constants present in the equation of the family of
curves.

**Example 4.2**

Find the differential equation of the family of straight lines y=mx+cwhen
(i) m is the arbitrary constant (ii) c is the arbitrary constant (iii) m and c both are arbitrary
constants.

*Solution:*

**Example 4.3**

Find the differential equation of the family of curves y= a/x + b
where a and b are arbitrary constants.

*Solution:*

**Example 4.4**

Find the differential equation corresponding to y = ae^{4x}
+ be^{−x} where a, b are arbitrary constants,

*Solution:*

**Example 4.5**

Find the differential equation of the family of curves *y* = *e ^{x}* (

*Solution :*

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12th Business Maths and Statistics : Chapter 4 : Differential Equations : Formation of ordinary differential equation | Example Solved Problems with Answer, Solution, Formula

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