(2) Show that xJo(x)dx = (x- 4x)/1(x)+2x](x)+c Hint: First we do integration by parts where u=x and dv=xJo(x)dx so we get du=2xdx and v=xJ1(x). (d)(20%) Now each candidate solution is given by v(r,t)=Jo(ar)exp(-ant) and clearly none of the candidate solutions can satisfy the initial condition v.(r,0)= Kb/4D*([(r/b)-1]. Let us propose that the unknown function v(r,t) can be built up using the eigenfunctions in (a). v(r,t) =Clo(at)exp(-Dat) Using the I.C., we can get Kb 2 v(r, 0) = f(r) = AD [(+) - 1] = CnJo (ant) n=1 Using the Sturm Louiville Theorem, show that we get Cm to be (Eqn 1.3-11) Cm 3 Ab So [b - 1] Jo (amr) dr 4D b2 Sr(amr)dr The integral on the denominator can be evaluated using Theorem 4 on page 813 to get b b(amb) rj (mr)dr = = [J (amb) + ] (amb)] = 2 The integral on the numerator can be evaluated to get Kb 3 AD [b = 1]10 (amr)dr = Kb] (amb) 4D -r Jo Dam Show that Cm is given by 2K Cm = Dba(an) Determine the transient solution v(r,t) and what is v(r,t)? Show that the average temperature Uave(t) defined by Uave (t) = * fru(r,t)dr is given by Kb 00 uave (t) + 8D _D,exp(-Dagt) What is the expression for Dn? (e)(20%) Let b=1, D=1 and K=4. Note that the an are the positive roots of the above equation. Write out a list of the first ten values for an, the eigenvalues, the corresponding eigenfunctions and the matching functions. For the same list of values for an, write out a list of Jo(an), Ji(an), Cn and Dn. Plot the temperature profile for times t = 0, 0.05, 0.1, 0.3, and infinity and provide the Vave for these times. In using the infinite series, ignore all terms with values less than 1 x10-10 and comment on the utility of using infinite series to get these answers.