Question

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

Hint:

To eliminate constants from a family of curves, differentiate the equation as many times as necessary and express the constants in terms of the variables.

Answer 1

Step 1: Differentiate with respect to \( x \).
To eliminate the constants \( a \) and \( b \), we differentiate the given equation twice. First, differentiate with respect to \( x \): \[ \frac{dy}{dx} = a \cos(x + b). \]
Step 2: Second Derivative.
Now, differentiate again: \[ \frac{d^2y}{dx^2} = -a \sin(x + b). \]
Step 3: Express \( a \) in terms of \( y \).
From the original equation \( y = a \sin(x + b) \), solve for \( a \): \[ a = \frac{y}{\sin(x + b)}. \]
Step 4: Substitute into the second derivative.
Substitute \( a \) into the second derivative equation: \[ \frac{d^2y}{dx^2} = -\frac{y}{\sin(x + b)} \sin(x + b). \] Simplifying, we get: \[ \frac{d^2y}{dx^2} = -y. \]
Step 5: Conclusion.
Thus, the differential equation of the family of curves is: \[ \frac{d^2y}{dx^2} + y = 0. \]