Take-home projectford? fferentialequat? ons 1. -)yl prelim? narywork look up the differential equation for a spring/mass system in section 5.1.2 of your book (equation (10) on p197) d2y dy use the same shorthand symbols from that section in your future work (formulas (12) on p197) and make sure you can follow the interpretation of cases i, il and i described on p198 (the figures 5.1.6, 5.1.7 and 5.1.8 should you understand the different possible motions of a mass attached to a spring) first, solve the (homogeneous) equation using the laplace transform technique. note that 40] = ? . you will have to distinguish the same three cases than on p198. your final answers should obviously match those on p198 then, imagine that the mass is given a very (infinitely) hard kick of extremely small (infinitesimal) duration at time to. the kick can be modeled by the "dirac function", whose laplace transform you have seen in the lecture. use the laplace transform technique to solve the non-homogeneous equation with the dirac function as the non-homogeneous term. you wil still have to distinguish case i, ll and ill. in a few words, interpret the resulting function y(t). 2. often integrals containing the exponential function with imaginary exponent are treated in the same way as if we were dealing with real exponentials, eg. in u-substitution, eixdx = eixd(ix) =-feud(u) = 1 eu + c. assume that this is justified (the proof would belong in a course in complex analysis).