The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. |
Home Exponential and Logartihmic Functions Some Differential Equations Examples Example 5 | |
Search the VIAS Library | Index | |
Example 5
When a spring of natural length L is compressed a distance x it exerts a force F = -kx. The negative sign indicates that the force is in the opposite direction from x (Figure 8.6.3). Figure 8.6.3 When x is negative the spring is expanded and the equation F = -kx still holds. Suppose a mass m is attached to the end of the spring and at time t = 0 is at position x0 and has velocity v0. The motion of the mass follows the differential equation The general solution is x = a cos cot + b sin ωt where Using the initial conditions, the motion of the mass is This function is periodic with period 2π/ω, so as expected the mass oscillates back and forth. In the following second order equation, hyperbolic sines and cosines arise. The general solution of the differential equation d2y/dx2 = y is y = a cosh x + b sinh x. We see that cosh x and sinh x are solutions because , , , . Another solution is ex. Note that
|
|
Home Exponential and Logartihmic Functions Some Differential Equations Examples Example 5 |