HW03
Assigned:      MON 01 FEB
Due:              FRI 12 FEB

Problem 1.   Barton Problem 2.1, p.67.

Problem 2.   Barton Problem 2.2, p.67.

Problem 3.   Barton Problem 2.3, p.67. Note that solving for the Green's function involves ONE equation and ONE unknown for a first order differential equation -- it is easier than the second order. Hence no determinate is required and the concept of the Wronskian becomes superfluous since there is now only one solution to the homogeneous problem and you need at least two for the notion of "two linearly independent solutions" to make sense.

Problem 4.   Barton Problem 2.4, p.68. Hint: This is a special (easier) case of the full Einstein solution presented in detail in Barton's Appendix D, so read over that carefully.  Note that solving for the Green's function involves ONE equation and ONE unknown for a first order differential equation -- it is easier than the second order. Hence no determinate is required and the concept of the Wronskian becomes superfluous since there is now only one solution to the homogeneous problem and you need at least two for the notion of "two linearly independent solutions" to make sense.

Problem 5.   Barton Problem 2.7, p.68. Typo: "...by the method of Section 2.2...." and not 2.3. This is an IVP not a BVP!

Problem 6.   Barton Problem 2.9, p.68. (Hint.) As in class, also provide a general expression for the kinetic energy as a function of time, and then analyze it in the limit that the driving frequency equals the -- damping shifted -- resonant frequency of the free oscillator and provide a description on the rate at which energy is transfered from the harmonic driving force into the damped oscillator. Finally, as in class, take the Fourier transform of the exact kinetic energy expression and analyze the results near resonance. (Hint: You should find that near resonance the resulting power spectrum becomes a Lorentzian, Barton Eq.1.2.4, centered about the resonance condition, which is now shifted slightly via the damping constant.) Show that as the damping goes to zero you recover the results in Problem 5, above.  Use the Lorentzian representation of the Dirac delta, also Eq.1.2.4. (Hint: Use Mathematica™! Also for the final Fourier transform be sure to use assumptions to force all time frequencies to be real and strictly positive. Otherwise you will get a lot of delta functions that are really all zero.)