- Books Name
- Mathmatics Book Based on NCERT
- Publication
- KRISHNA PUBLICATIONS
- Course
- CBSE Class 12
- Subject
- Mathmatics
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',
The required differential equation is then obtained by eliminating a and b from equations (4), (5) and (6) as
F (x, y, y',
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.