**Example:** Consider a harmonic oscillator for which the differential equation is

,

and suppose that mass *m*=1, the damping constant , and the spring constant . Rewrite this equation as a linear system of differential equations. Solve it, then find the particular solution which satisfies the initial conditions

**Answer.** Set *v*=*y*'. Then we have

This gives us the system

which in matrix form may be rewritten as

where

In order to solve this system, we need the characteristic equation

Its roots are given by the quadratic formulas

Note that you have to be very careful here since any mistake at finding correctly the roots will generate a far bigger mistakes and waist of time!!

Next we need to find the associated eigenvectors.

- Case . Denote by the associated eigenvector. The system giving is
Since

(which you should check as an exercise), then the two equations are identical. Hence we take . If we choose , we get

- Case . Similar calculations give the associated eigenvector

where and are two parameters.

From the above equation giving *Y*, we may find the solution *y* to our second differential equation as

We are almost done except that we need to find the specific solution which satisfies the initial condition

These two conditions imply

The second equation gives

since , we get which implies . Hence we have

which implies

and

which yields

**
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