Real Eigenvalues: Example1

Example: Consider the harmonic oscillator with spring constant , damping constant , and the mass m=1.

(1)

Write down the second order equation governing this physical system. Use the letter y for the spring's displacement from its rest position.

(2)

Convert this equation into a linear system of first order differential equations.

(3)

Solve the system.

(4)

Find the particular solution which satisfies the initial conditions

(5)

Discuss the long-term behavior of the system. Is this conclusion probable?

Solution:

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