Problem

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Consider the position function .

Part 1: Determine the value of if this function solves the differential equation:

.

Part 2: Try to explain what each term of the above differential equation means.

Solution

Part 1

Take two time derivatives:

.

Consequently,

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Divide away :

.

Therefore .

This quantity is the angular frequency of the wave. Mathematically speaking, it is the eigenvalue of the differential equation.

Part 2

Multiply the entire equation by :

.

Since acceleration is the second time derivative of position

.

This is Newton’s second law with the spring force being the net force!

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