Example 3.  Find the general solution to the system of D. E.'s  and plot the solution curves.

Solution 3.

Put the D.E.'s in operator form and eliminate y to obtain a fourth order D.E. for x, and find the roots of its characteristic equation.

The roots are pure complex,  , and the natural frequencies  ,  respectively.  The general solution is formed as follows.

Aside.  The eigenfrequencies can be obtained by taking the square root of the eigenvalues of the matrix .

It is useful to look at the two natural modes of oscillation of the spring mass system and they exhibit the natural frequencies  ,  respectively.

We can verify these solutions.  

Plot the functions   and  .  In this mode of oscillation the masses are moving in opposite directions.

Plot the functions   and  .  In this mode of oscillation the masses are moving in the same directions.

Assume that the equilibrium position along the horizontal axis is 2 and 6.  The two masses move in the opposite direction with the frequency  , as seen in the next graph, where time is along the vertical axis.

Assume that the equilibrium position along the horizontal axis is 2 and 6.  The two masses move in same directions with the frequency , as seen in the next graph, where time is along the vertical axis.

(c) John H. Mathews 2005