Need some help with both parts. Please show work. Thank you! This is an example

Question

Need some help with both parts. Please show work.

Thank you!

This is an example of an Undamped Forced Oscillation where the phenomenon of Beats Occurs. Find the solution of the initial value problem: x" + 33.64X = 2cos(6t), x(0) = x' (0) = 0 x{t) = Graph the solution to confirm the phenomenon of Beats. Note that you may have to use a large window in order to see more than one beat. What is the length of each beat? Length = Would you be able to explain why the beats phenomenon occurs for this particular example?

Explanation / Answer

given equation

x '' +33.64 x= 2cos(6t), x(0)=x '(0)=0

the auxilary equation is

m2 + 33.64=0

m= + - 5.8i

xc(t)= c1 cos 5.8t + c2 sin 5.8t

for particular solution,

xp(t)= A cos6t + Bsin6t

x 'p= -6A sin6t + 6B cos6t

x ''p=-36A cos6t - 36B sin6t

replacing in question

x '' +33.64 x= 2cos(6t)

36A cos6t - 36B sin6t + 33.64(A cos6t + Bsin6t)= 2cos (6t)

comparing coefficients of cos 6t and sin 6t,

36A -33.64A =2, 2.36A = 2, A= 0.847

-36B+ 33.64B =0   , B=0

x= c1 cos 5.8t + c2 sin 5.8t + 0.847cos6t

x '(t)=-5.8c1sin 5.8t + 5.8c2cos 5.8t + 0

x(0)=x '(0)=0

0=c1+ 8.847

c1= -0.847

0=5.8c2

c2=0

x= -0.847 cos 5.8t + 0.847cos6t

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