Question

2.
Form the differential equation representing the family of curves y = A cos(x + B),
Where A and B are parameter.​

Answer 1

Basic Concept Used :-

To form a differential equation of family of curves, the following steps have to be followed :-

• 1. Count the number of arbitrary constants in the given curve, say n.

• 2. Differentiate the given curve 'n' times to get 'n + 1' equations.

• 3. Eliminating the arbitrary constants with the help of these 'n + 1' equations.

• 4. Rest is the required differential equation.

Solution :-

Given family of curve is

$\red{\bf :\longmapsto\:y = a \: cos(x + b)} - - - (1)$

• where a and b are parameter or arbitrary constants.

So,

It means we have to differentiating the given curve twice to obtain the required differential equation.

Now, Differentiate equation (1) w. r. t. x, we get

$\green{\bf :\longmapsto\:y_1 = - a \: sin(x + b)} - - - (2)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \because{ \bf \: \dfrac{d}{dx}cosx = - \: sinx}}$

Again, On differentiating equation (2) w. r. t. x, we get

${\bf :\longmapsto\:y_2 = - a \: cos(x + b)}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \because{ \bf \: \dfrac{d}{dx}sinx = \: cosx}}$

$\purple{{\bf :\longmapsto\:y_2 = - \: y \: \: \: \: \{using \: (1) \}}}$

$\purple{{\bf :\implies\:y_2 + y \: = \: 0\: is \: required \: differential \: {eq}^{n}}}$