 form a diff eqn from x = A cos ( p t - α ) 

The given equation is \( x = A \cos(p t - \alpha) \), which describes a harmonic oscillator, where:

- \( A \) is the amplitude,
- \( p \) is the angular frequency, and
- \( \alpha \) is the phase shift.

To form a differential equation from this, we differentiate with respect to time (\( t \)) and eliminate the constants to obtain a relationship involving \( x \) and its derivatives.

### Step 1: First Derivative
The first derivative of \( x \) with respect to time gives the velocity:
\[
\frac{dx}{dt} = \frac{d}{dt} \left( A \cos(p t - \alpha) \right) = -A p \sin(p t - \alpha)
\]

### Step 2: Second Derivative
The second derivative of \( x \) with respect to time gives the acceleration:
\[
\frac{d^2x}{dt^2} = \frac{d}{dt} \left( -A p \sin(p t - \alpha) \right) = -A p^2 \cos(p t - \alpha)
\]

Since \( x = A \cos(p t - \alpha) \), we can substitute \( x \) back into this second derivative equation:

\[
\frac{d^2x}{dt^2} = -p^2 x
\]

### Step 3: Final Differential Equation
The resulting differential equation is:

\[
\frac{d^2x}{dt^2} + p^2 x = 0
\]

This is a second-order linear homogeneous differential equation, characteristic of simple harmonic motion.

Would you like more details or further explanation?

Here are 5 related questions you could explore:
1. How does the solution of this differential equation describe simple harmonic motion?
2. What is the physical meaning of the angular frequency \( p \) in this context?
3. How would the equation change if there was damping involved in the system?
4. What initial conditions are necessary to fully determine the constants \( A \) and \( \alpha \)?
5. How would you solve this equation for different boundary conditions?

**Tip:** The general form of a simple harmonic motion equation is always of the type \( \frac{d^2x}{dt^2} + \omega^2 x = 0 \), where \( \omega \) is the angular frequency.

Form a differential equation from x = A cos(p t - α).

Learn how to form a second-order linear homogeneous differential equation from the harmonic oscillator equation x = A cos(p t - α). Explore key concepts in differential equations and simple harmonic motion, with step-by-step differentiation and substitution techniques.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation