A forced damped harmonic oscillator satisfies the following equation. What is th

Question

A forced damped harmonic oscillator satisfies the following equation. What is the
general solution x(t)?

2d2x/dt2 + 8dx/dt + 6x = 4cos(5t)

When determining the particular solution and the homogeneous solution, do not just
substitute into a formula for amplitude A, etc. Assume a trial function for each
solution and work out the parameters that you need to find "from scratch." Assume
a trial function of the form xH = AH eqt for the homogeneous solution, and a function
of the form xP = AP ei (t – ø) for the particular solution. (By inspection of the
differential equation, you should know what equals in this problem.)

Express your final answer as a real function, that is, there should be no i’s in your final answer
(where i = (1)½). In your particular solution, the constants that you have
determined should be correct to two significant digits (except for , for which one
significant digit is fine).

Explanation / Answer

let x=A*e^i(wt-0). x'=A*iw*e^i(wt-0). x''=-A*w^2*e^i(wt-0). 2d2x/dt2 + 8dx/dt + 6x = 4cos(5t) so (2*(-A*w^2)+8*iw*A+6*A)*e^i(wt-0) (-2*A*w^2+6A+i8Aw)*(cos(wt-0)+i*sin(wt-0)). (6A-2Aw^2)*cos(wt-0)-8Aw*sin(wt-0)=4cos(5t) so w=5. so it is -46A*cos(wt-0)-40A*sin(wt-0)=4cos(5t). so sqrt(46^2+40^2)*A=4. so A=0.07. at t=0 we have. -46A*cos(0)+40A*sin(0)=4. so -3.22*cos(0)+2.8*sin(0)=4. solve this to get 0

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