Question

Find the differential equation of the family of curves denoted by \(y = a \sin(x+b)\), where a and b are arbitrary constants.

Hint:

The number of arbitrary constants in the general solution of a differential equation determines the order of the differential equation. Two constants mean you'll need to differentiate twice to eliminate them, resulting in a second-order equation.

Answer 1

Step 1: Understanding the Concept:
To form a differential equation from a general solution that contains 'n' arbitrary constants, we must differentiate the equation 'n' times. This creates a system of 'n+1' equations (the original plus the 'n' derivatives). The goal is to algebraically manipulate these equations to eliminate all 'n' arbitrary constants.
Step 2: Key Formula or Approach:
The given equation has two arbitrary constants, 'a' and 'b'. Therefore, we need to differentiate the equation two times to obtain a second-order differential equation free of 'a' and 'b'.
Step 3: Detailed Explanation:
The given equation for the family of curves is: \[ y = a \sin(x+b) \quad \text{---(1)} \] Differentiate equation (1) with respect to x: \[ \frac{dy}{dx} = \frac{d}{dx}[a \sin(x+b)] \] Using the chain rule: \[ \frac{dy}{dx} = a \cos(x+b) \cdot \frac{d}{dx}(x+b) = a \cos(x+b) \quad \text{---(2)} \] Since the constants are not yet eliminated, we differentiate again with respect to x: \[ \frac{d^2y}{dx^2} = \frac{d}{dx}[a \cos(x+b)] \] Using the chain rule again: \[ \frac{d^2y}{dx^2} = -a \sin(x+b) \cdot \frac{d}{dx}(x+b) = -a \sin(x+b) \quad \text{---(3)} \] Now we have three equations. We can eliminate the constants 'a' and 'b' by comparing equations (1) and (3). From equation (1), we have \(y = a \sin(x+b)\). From equation (3), we have \(\frac{d^2y}{dx^2} = -[a \sin(x+b)]\). By substituting the expression for y from (1) into (3), we get: \[ \frac{d^2y}{dx^2} = -y \] Rearranging the terms to form the standard differential equation format: \[ \frac{d^2y}{dx^2} + y = 0 \] This is the required differential equation, as it is free from the arbitrary constants 'a' and 'b'.
Step 4: Final Answer:
The differential equation of the family of curves is \(\frac{d^2y}{dx^2} + y = 0\).