Formation of Differential Equations

Consider a family of curves

f( x , y , α1 , α2 , …. , αn ) = 0 . . . . . (1)

Where α1, α2, …..,αn are n independent parameters.

Equation (1) is known as an n parameter family of curves

e.g. y = mx is a 1 – parameter family of straight lines ,

x2 + y2 + ax + by = 0 is a two-parameters family of circles.

If we differentiate equation (1) n times w.r.t. x , we will get n more relations between x , y , α1, α2 , …., αn and derivatives of y w.r.t. x .

By eliminating α1 , α2 , …., αn from these n relations and equation (1), we get a differential equation.

Clearly order of this differential equation will be n i.e. equal to the number of independent parameters in the family of curves.

e.g. consider the family of parabolas with vertex at the origin and axis as the x-axis

y2 = 4ax . . . . . (1)

Differentiating w. r. t. x , we get

$ \displaystyle 2 y \frac{dy}{dx} = 4 a $

$ \displaystyle 2 y \frac{dy}{dx} = \frac{y^2}{x} $ from (1)

Or , $ \displaystyle 2 x \frac{dy}{dx} – y = 0 $

which is the differential equation of (1) and is clearly of order 1 .

Example : Find the differential equation whose solution represents the family
y = A sin x + B cos x + x2

Solution: y = A sin x + B cos x + x2 … (1)

Differentiating (1) with respect to x , we get

$ \displaystyle \frac{dy}{dx} = A cosx – B sinx + 2 x $ …(2)

Differentiating (2) with respect to x, we get

$ \displaystyle \frac{d^2 y}{dx^2} = – A sinx – B cosx + 2 $

$ \displaystyle \frac{d^2 y}{dx^2} = – (A sinx + B cosx )+ 2 $

$ \displaystyle \frac{d^2 y}{dx^2} = – (y – x^2 ) + 2 $ (from (1))

$ \displaystyle \frac{d^2 y}{dx^2} + y – x^2 = 2 $

which is the required differential equation.

Remark:

∎ The order of the differential equation will be equal to the number of independent parameters and not equal to the number of all the parameters in the family of curves.

e. g consider the family of the curves y = (C1 + C2 ) ex + C3

Here the number of arbitrary parameters is 4 but the order of the corresponding differential equation will not be 4 as it can be rewritten as

y = (C1 + C2 + C3 ) ex , which is of the form y = A ex . Hence the corresponding differential equation will be of order 1.

Exercise :

(i) Show that the differential equation of the family of circles of fixed radius r with centres on y−axis is
$ \displaystyle (x^2 -r^2) (\frac{dy}{dx})^2 + x^2 = 0 $

(ii) Find the order of the following curves:

(a) y = c1 tan2 x + c2 sec2 x + c3

(b) Family of straight lines passing through origin.

(iii) Find the differential equation of all non-vertical lines in a plane.

(iv) Find the differential equation of all circles touching the y-axis at the origin.

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