Consider the vibrating system described by the initial value problem. (A computer algebra system is recommended.) u'' + u = 8 cos ωt, u(0) = 5, u'(0) = 7 (a) Find the solution for ω ≠ 1.

Step-by-step explanation:

We have the differential equation

This differential equation can be solved by the undetermined coefficient method. The general solution is

u(t) = uh(t) + up(t)

where uh(t) is the solution to the homogeneous equation and up(t) is a particular solution

1. First of all, we calculate uh(t) by using the characteristic polynomial:

The solution are imaginary:

r1=+i

r2=-i

hence, uh(t) is

where A and B are constant that we have to calculate.

2. Now we calculate up(t). In the undetermined coefficient method we can assume that the solution is up(t)=Ccos(wt)+Dsin(wt). By taking derivatives of up(t) we have

and by replacing in the differential equation

Hence we have

where we have taken D=0 because it is clear that the solution does not depended of sin functions.

3. Finally the u(t) is

and by applying the initial conditions we have

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