The following pertains to ordinary differnetial equations OD

The following pertains to ordinary differnetial equations (ODE\'s):

a) A mass-spring oscillator (m = 1 and k = 4) has the following equation of motion:

x\'\' + 4x = 0

Solve for the response variable x(t), using the following initial conditions: x(0) = 5, x\'(0) = 0.

Solution

The mass spring oscillator is the case of simple harmonic motion for which the response is given by

x(t) = A sin(Bt+C) for the equation x\" + B²x = 0

Given equation is x\" + 4x = 0.

===> B = 4 = 2

===> x(t) = A sin(2t+C) ---- (1)

At t = 0, x = 5 ===> A sin C = 5 ---- (2)

Upon differentiating equation (1) with reference to t, we get

x\'(t) = 2A cos (2t+C) ---- (3)

At t = 0, x\'(0) = 0 ===> 2A cos C = 0

===> C = /2

From (2), we get A sin /2 = 5 ===> A = 5

From (1), we get x(t) = 5 sin(2t+/2)

===> x(t) = 5 cos 2t

This is the solution for the response variable x(t)