Example 1.  Equal Masses, Unequal spring constants.  Find the general solution to the system of D. E.'s  and plot the solution curves.

Solution 1.

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.

Aside.

We could plot infinitely many types of oscillation.  For illustration we plot the solution for the coefficients .  

(c) John H. Mathews 2005