(*^ ::[ fontset = title, "Helv", 22, L0, center, nohscroll, bold; fontset = subtitle, "Helv", 18, L0, center, nohscroll, bold; fontset = subsubtitle, "Helv", 12, L0, center, nohscroll, bold; fontset = section, "Helv", 12, L0, bold, grayBox; fontset = subsection, "Helv", 12, L0, bold, blackBox; fontset = subsubsection, "Helv", 10, L0, bold, whiteBox; fontset = text, "Helv", 12, L0; fontset = smalltext, "Helv", 10, L0; fontset = input, "Courier", 12, L0, nowordwrap; fontset = output, "Courier", 12, L0, nowordwrap; fontset = message, "Courier", 10, L0, nowordwrap, R65280; fontset = print, "Courier", 10, L0, nowordwrap; fontset = info, "Courier", 10, L0, nowordwrap; fontset = postscript, "Courier", 10, L0, nowordwrap; fontset = name, "Helv", 10, L0, nohscroll, italic, B65280; fontset = header, "Helv", 18, L0, nohscroll, bold; fontset = footer, "Helv", 18, L0, center, nohscroll, bold; fontset = help, "Helv", 10, L0, nohscroll; fontset = clipboard, "Helv", 12, L0, nohscroll; fontset = completions, "Helv", 12, L0, nowordwrap, nohscroll; fontset = network, "Courier", 10, L0, nowordwrap, nohscroll; fontset = graphlabel, "Courier", 10, L0, nowordwrap, nohscroll; fontset = special1, "Helv", 12, L0, nowordwrap, nohscroll; fontset = special2, "Helv", 12, L0, center, nowordwrap, nohscroll; fontset = special3, "Helv", 12, L0, right, nowordwrap, nohscroll; fontset = special4, "Helv", 12, L0, nowordwrap, nohscroll; fontset = special5, "Helv", 12, L0, nowordwrap, nohscroll; fontset = Left Header, "Helv", 12, L0, nowordwrap, nohscroll; fontset = Left Footer, "Helv", 12, L0, nowordwrap, nohscroll;] :[font = subsection; inactive; ] Example 9. :[font = text; inactive; ] Find a power series solution of the following initial value problem: x''(t)+4 x(t)=0; x(0)=1 and x'(0)=0. The coefficients of this equation are analytic everywhere and in paricular at 0. Thus we expect to find a power series solution of the form: . Our task is to find these coefficients. Denote them by a[n]. Let x[a,t,k] denote the k-th partial sum of the power series solution. Substitute it into the given equation and collect the like powers of t. Below, it is done for k=8. :[font = input; startGroup; ] x[a_,t_,k_]=Sum[(a[n] t^n),{n,0,k}] Cs=Collect[D[x[a,t,8],t,t]+4 x[a,t,8],t] :[font = output; inactive; formatted; output; ] Sum[t^n*a[n], {n, 0, k}] ;[o] n Sum[t a[n], {n, 0, k}] :[font = output; inactive; formatted; output; endGroup; ] 4*a[0] + 2*a[2] + t*(4*a[1] + 6*a[3]) + t^2*(4*a[2] + 12*a[4]) + t^3*(4*a[3] + 20*a[5]) + t^4*(4*a[4] + 30*a[6]) + 4*t^7*a[7] + t^5*(4*a[5] + 42*a[7]) + 4*t^8*a[8] + t^6*(4*a[6] + 56*a[8]) ;[o] 4 a[0] + 2 a[2] + t (4 a[1] + 6 a[3]) + 2 3 t (4 a[2] + 12 a[4]) + t (4 a[3] + 20 a[5]) + 4 7 5 t (4 a[4] + 30 a[6]) + 4 t a[7] + t (4 a[5] + 42 a[7]) + 8 6 4 t a[8] + t (4 a[6] + 56 a[8]) :[font = text; inactive; ] Now, we set the coefficients of t^n equal to zero, say for n=1,2,...6. For clarity, we put them in the form of a table as follows: :[font = input; startGroup; ] TableForm[Table[{p,Coefficient[Cs,t,p]==0},{p,0,6}]] :[font = output; inactive; formatted; output; endGroup; ] TableForm[{{0, 4*a[0] + 2*a[2] == 0}, {1, 4*a[1] + 6*a[3] == 0}, {2, 4*a[2] + 12*a[4] == 0}, {3, 4*a[3] + 20*a[5] == 0}, {4, 4*a[4] + 30*a[6] == 0}, {5, 4*a[5] + 42*a[7] == 0}, {6, 4*a[6] + 56*a[8] == 0}}] ;[o] 0 4 a[0] + 2 a[2] == 0 1 4 a[1] + 6 a[3] == 0 2 4 a[2] + 12 a[4] == 0 3 4 a[3] + 20 a[5] == 0 4 4 a[4] + 30 a[6] == 0 5 4 a[5] + 42 a[7] == 0 6 4 a[6] + 56 a[8] == 0 :[font = text; inactive; ] Next we need to conjecture a recurrence relation for a[n] and we are able to compute a[0] and a[1]. In fact, since x(0)=1 and x'(0)=0, then a[0]=1 and a[1]=0. Furthermore, the recurrence relation is: a[n]= -4 a[n-2]/(n(n-1)). That conjecture is entered below to define the coefficient function ai[j]: :[font = input; ] Clear[ai] :[font = input; startGroup; ] ai[0]=1 ai[1]=0 ai[j_]:=-4 ai[j-2]/(j(j-1)) :[font = output; inactive; formatted; output; ] 1 ;[o] 1 :[font = output; inactive; formatted; output; endGroup; ] 0 ;[o] 0 :[font = text; inactive; ] Now we are in a position to find all the coefficients: :[font = input; startGroup; ] Table[ai[j],{j,20}] :[font = output; inactive; formatted; output; endGroup; ] {0, -2, 0, 2/3, 0, -4/45, 0, 2/315, 0, -4/14175, 0, 4/467775, 0, -8/42567525, 0, 2/638512875, 0, -4/97692469875, 0, 4/9280784638125} ;[o] 2 4 2 4 4 {0, -2, 0, -, 0, -(--), 0, ---, 0, -(-----), 0, ------, 0, 3 45 315 14175 467775 8 2 4 -(--------), 0, ---------, 0, -(-----------), 0, 42567525 638512875 97692469875 4 -------------} 9280784638125 :[font = input; startGroup; ] xi[t_,k_]:=x[ai,t,k] xi[t,20] :[font = output; inactive; formatted; output; endGroup; ] 1 - 2*t^2 + (2*t^4)/3 - (4*t^6)/45 + (2*t^8)/315 - (4*t^10)/14175 + (4*t^12)/467775 - (8*t^14)/42567525 + (2*t^16)/638512875 - (4*t^18)/97692469875 + (4*t^20)/9280784638125 ;[o] 4 6 8 10 12 14 2 2 t 4 t 2 t 4 t 4 t 8 t 1 - 2 t + ---- - ---- + ---- - ----- + ------ - -------- + 3 45 315 14175 467775 42567525 16 18 20 2 t 4 t 4 t --------- - ----------- + ------------- 638512875 97692469875 9280784638125 :[font = text; inactive; ] Observe now that if we use Mathematica to solve the differential equation, then the solution is Cos(2t) for which the first twenty terms of its Maclaurin Series is exactly what we have obtained above. 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We would get a better approximation over a larger interval by taking more terms of the power series of Cos(2t). However no matter how many terms are used, the approximation will be very poor outside some finite interval [-s,s] because the exact solution is bounded between -1 and 1 for all t, whereas the partial sum x[t,k] is a polynomial which must be unbounded. :[font = text; inactive; ] :[font = input; locked; startGroup; ] TableForm[ai[t],{t.11,20}] :[font = message; inactive; formatted; output; ] Can't Format Cell ;[o] $RecursionLimit::reclim: Recursion depth of 256 exceeded. :[font = message; inactive; formatted; output; ] Can't Format Cell ;[o] $RecursionLimit::reclim: Recursion depth of 256 exceeded. :[font = message; inactive; formatted; output; ] Can't Format Cell ;[o] $RecursionLimit::reclim: Recursion depth of 256 exceeded. :[font = message; inactive; formatted; output; endGroup; ] Can't Format Cell ;[o] General::stop: Further output of $RecursionLimit::reclim will be suppressed during this calculation. :[font = input; locked; ] ai[11] :[font = input; ] ^*)