2. FORCED DAMPED HARMONIC MOTION In the physical world damping is always present, thus we should consider what happens when we add some damping to our harmonic oscillator model. This is done by adding a term ca' where c is a constant, a" + or + wor = A cos(wt) (6) Consider the nonhomogenous differential initial value problem 0.2p" + 1.2x' + 2x -5 cos(4t) (0) = 0.5, r'(0) = 0. If the above IVP is solved, then the complete set of terms of which the solution is comprised are listed below 25 (a) 105 Cos( 4t) 50 (b) sin(47) (c) -e-3 56 sin(() (d) e- at 38 cos(1) Answer the following questions about the above differential equation and its solutions. (a) What is the homogeneous solution, ya, also called the complementary solutions, y? (b) What is the particular solution, y,? (c) If a (t) represents the complete solution of the IVP, what is r(t)? (d) If e( t) represents the complete solution of the IVP, what is lim, a(t)? (e) What are the transient terms, and do they come from ya or y? POINT OF INTEREST: Large transient currents (currents that die out after a short time) can be destructive in electrical circuits. For example, a light bulb usually burns out when a large transient Hows through an already weakened bulb. This, usually occurs at the time the light bulb is turned out or turned off. Large transient currents are particularly harmful to the hard drive in a computer. For this reason experts advise that computers should be left on except in the case that they are not being used for a period of several days. O' T 99+