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