Formation of a Differential Equation

Consider the family of lines represented by y = mx ….(1)Consider the family of lines represented by y = mx ….(1)

This equation represents infinite number of lines passing through the origin.

Differentiating (1), we get

Substituting this value of m, we get the differential equation.

Consider the family of lines represented by

y = mx + c ….(3)

where m and c are arbitrary constants. Any line on the co-ordinate plane can be represented by (3)

Let us form a differential equation for equation (3)

Differentiating equation (3) , we have

Differentiating again, we have

This is the differential equation which represents the family of straight lines y = mx +c.

Note:

The equation y = mx has one arbitrary constant and its differential equation is of order 1. The equation y = mx + c has two arbitrary constants and its differential equation is of order 2.

In general, if an equation contains n arbitrary constants, then we obtain its differential equation which is of order n, after eliminatory all the n constants.

Note that equation(1) is called the primitive of Differentiating equation(2).

Equation (3) is called the primitive of Differentiating equation (4).

The formation of a differential equation may be done by differentiating and eliminating arbitrary constants from the given equation.

The given equation is differentiated as many times as there are arbitrary constants.

Formation of the Differential Equation that will Represent a Given Family of Curves

In this section we explained with examples how to form Differential Equation.

Suppose a family of curves depending on one constant is given by

F₁ : f (x, y, a)= 0 ….(1)

where a R is the parameter

Differentiating (1) with respect to x, we have

g (x, y, y', a) = 0 ….(2)

Now eliminating 'a' from equation (1) and equation (2), we get the required differential equation.

f (x, y, y') = 0 ….(3)

This equation represents the family of curves F₁. Equation (1) is called the primitive of the differential equation (3).

In this section, we discuss in general how to form Differential Equation which represent a family of curves.

Let

F₂: f (x, y, a, b) = 0 ….(4)

where a, b R

represents a family of curves which depend on two constants (parameters) a, b.

Differentiating (4) with respect to x, we have

g (x, y, y', a, b) = 0 ….(5)

We can not eliminate a and b from equation (4) and equation (5). Therefore we need another equation which can be obtained by differentiation equation (5).

Differentiating equation (5) with respect to x, we have

h(x, y, y ',y'', a, b) = 0 ….(6)

Now the arbitrary constants can be eliminated from equation (4), (5) and (6), to obtain the differential equation of the family of curves.

As discussed earlier, the family of curves containing one parameter, is represented by a differential equation of order 1 .

The family of curves which depend on two parameter is represented by differential equation of order 2.

In general, the family of curves which depend on n parameter is represented by differential equation of order 3.

Example 1:

Form a differential equations by eliminating 'a' from the family of curves y²=4ax.

Suggested answer:

y² = 4ax …(1)

Differentiating with respect to x

Substitute for 4a in (1), we get

y - 2xy' = 0

Note that the given equation is differentiated only once to obtain the differential equation since it has only one constant.

Example 2:

Form a differential equation by eliminatory the parameter A and B from the family of curves given by y = Ae^2x+Be^-2x.

Suggested answer:

The given equation has two arbitrary constants.

\ To obtain the differential equation, we differentiate the given equation twice.

Differentiate with respect to x.

y₂ = 4y

y₂ - 4y = 0 which is the required differential equation.