4. Formation of a Differential Equation whose General Solution is given

Formation of a Differential Equation whose General Solution is given

The order of a differential equation representing a family of curves is the same as the number of arbitrary constants present in the equation corresponding to the family of curves.

Procedure to form a differential equation that will represent a given

family of curves:

(a) If the given family F1 of curves depends on only one parameter then it is

represented by an equation of the form

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

For example, the family of parabolas y2 = ax can be represented by an equation of the form

 f (x, y, a) : y² = ax.

Differentiating equation (1) with respect to x, we get an equation involving y', y, x, and a,

i.e.,

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

The required differential equation is then obtained by eliminating a from equations (1) and (2) as

F(x, y, y' )  = 0 ..................... (3)

(b) If the given family F2 of curves depends on the parameters a, b (say) then it is represented by an equation of the from

F2 (x, y, a, b) = 0  …………………….(4)

Differentiating equation (4) with respect to x, we get an equation involving , x, y, a, b,

i.e.,

g (x, y, y'', a, b) = 0   ………………………(7)

But it is not possible to eliminate two parameters a and b from the two equations and so, we need a third equation. This equation is obtained by differentiating equation (5), with respect to x, to obtain a relation of the form

h (x, y, y', y'', a, b) = 0 .......................... (6)

The required differential equation is then obtained by eliminating a and b from equations (4), (5) and (6) as

F (x, y, y', y'')= 0 .................................. (7)

Note: The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding to the family of curves.

 

Example : Form the differential equation representing the family of curves y = mx,

where, m is arbitrary constant.

Solution: We have

y = mx ... …………………(1)

Differentiating both sides of equation (1) with respect to x, we get

y’ = m

Substituting the value of m in equation (1) we get

            y= y’ x

• xy’-y=0

which is free from the parameter m and hence this is the required differential equation.

Example : Form the differential equation representing the family of curves

y = a sin (x + b), where a, b are arbitrary constants.

Solution :We have

y = a sin (x + b) ... ……………(1)

Differentiating both sides of equation (1) with respect to x, successively we get

y’ =a cos(x+b)…………………..(2)

• y’’ = -a sin(x+b)………………….(3)

Eliminating a and b from equations (1), (2) and (3), we get

 y’’ +y =0    ………………………(4)

which is free from the arbitrary constants a and b and hence this the required differential equation.