Chapter 1: Problems 19 2. For what real number(s) k is the function f(x, t) = cos t k ekx a solution to the differential equation fx + ftt = 0? 3. (a) Use the Mathematica animation techniques from Section 1.6.1 to make a “movie” illustrating the dynamics of the function f(x, y) = 2e−(x+t)2 + 1 on the viewing window −10 x 10 and 0 y 3 for −10 t 10 with fifteen frames. The technical mathematical term for how this solution changes is “translation”. How would you describe it in nontechnical terms? (Hint: You can refer to ex in Mathematica either as E^x (with a capital “E”) or Exp[x]. See Appendix A for more advice on using this software.) (b) Use the Mathematica animation techniques from Section 1.6.1 to make a “movie” illustrating the dynamics of the function u(x, y, t) = sin(x2 + y2 t) x2 + y2 + 1 on the viewing window −4 x 4, −4 y 4 and −1 z 1 for 0 t 2π. What phenomenon that you might see on the surface of a pond does it resemble? 4. Suppose that u(x, t) = f(x + αt) is a solution to some partial differential equation where f(z) is some function of one variable and α is some constant. (a) What does the initial profile look like? (b) How does the solution change as time passes? (If you made an animation, what would it look like?) (c) What is the significance of the sign of the parameter α? (d) What is the significance of the absolute value of the parameter α? 5. For each fixed value of the parameter k, the function7 fk(x, t) = sech(x2 +tk) depends on one spatial and one temporal variable. In Mathematica, define f[x_,t_,k_]:=Sech[x^2+t^k] and use the MyAnimate command to compare the dynamics for the values k = 7 The hyperbolic secant function is defined as sech(z) = 2/(ez + e−z).
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