Real Eigenvalues: Answer to Example1

Example: Consider the harmonic oscillator with spring constant tex2html_wrap_inline96 , damping constant tex2html_wrap_inline98, 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

displaymath104

(5)
Discuss the long-term behavior of the system. Was this conclusion probable?

Answer:

(1)
The differential equation is

displaymath106.

Using the values for the constants, we get

displaymath108.

(2)
Set y'=v, then we have

displaymath112.

Hence, we have the system

displaymath114;

(3)
In order to solve the above system, we first need to find the eigenvalues of the system. Note that the matrix coefficient is

displaymath116.

The characteristic equation is given by

displaymath118.

Its roots are

displaymath120,

which gives

displaymath122

For every eigenvalue, we need to find an eigenvector.

Now we are ready to write down the general solution

displaymath156,

where

displaymath158.

(4)
In order to find the solution to the harmonic oscillator system which satisfies the initial conditions y(0)= 0 and y'(0)=1, we need the general solution which gives y. From the general solution to the system we get

displaymath166,

and tex2html_wrap_inline168 . The equation giving v is obvious and can be obtained from y since v=y' (you may want to check that we did not make any mistakes). The initial conditions imply

displaymath176

Solving it we get

displaymath178.

Therefore, the solution is

displaymath180

(5)
The long-term behavior of the solution is now obvious since

displaymath182,

meaning that the system tends to its rest position. Note that since the eigenvalues are both negative, it was clear from the outset that the solution will tend to its unique equilibrium position.

Next Example:

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Author: Mohamed Amine Khamsi

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