MATH 400 Midterm: MATH 400 2016 Winter Test 2
Document Summary
Answer as much as you can; credit awarded for the best three answers. Adequately explain the steps you take. e. g. if you use an expansion formula, say in one sentence why this is possible; if you quote a special function solution to an ode, say why this is the correct one. Be as explicit as possible in giving your solutions: using separation of variables, solve the wave equation, (cid:18)sin inside the unit sphere, r 1, with the boundary condition, u = 0 on r = 1, (cid:19) = utt, and initial condition, u(r, , 0) = 0 ut(r, , 0) = cos3 g(r). Hint: for the radial part of the problem, the substitution r(r) = x(r)/ r, may prove useful, if one sets u(r, , t) = r(r)y ( )t (t): establish that f g = f 1{ f g}, (1 + x2) where f{f} = f (k), f{g} = g(k), f g is a convolution, and a > 0.
