{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle1" -1 211 "Times" 1 18 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle2" -1 212 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 213 "Times" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle1" -1 201 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle2" -1 202 1 {CSTYLE "" -1 -1 " " 1 18 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }3 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 } {PSTYLE "_pstyle3" -1 203 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle 4" -1 204 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 202 "" 0 "" {TEXT 211 58 "Module 13: Review of Elementary Differential Equations II" }} {PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 212 212 " \+ In order that ideas from ordinary differential equations will be fre sh when we begin a discussion of partial differential equations, we co ntinue our review of some ideas from ordinary differential equations. " }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 212 48 " We make our review by asking six questions." }}{PARA 201 "" 0 "" {TEXT 213 11 "Question 1." }{TEXT 212 9 " Suppose " }{XPPEDIT 18 0 "la mbda;" "6#%'lambdaG" }{TEXT 212 76 " > 0. Which of these is a pair of \+ linearly independent solutions for Y '' - " }{XPPEDIT 18 0 "lambda^2; " "6#*$%'lambdaG\"\"#" }{TEXT 212 14 " Y = 0 on [0, " }{XPPEDIT 18 0 " pi;" "6#%#piG" }{TEXT 212 2 "]?" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }} {PARA 201 "" 0 "" {TEXT 212 7 "a. exp(" }{XPPEDIT 18 0 "lambda;" "6#%' lambdaG" }{TEXT 212 14 " x) and exp( -" }{XPPEDIT 18 0 "lambda;" "6#%' lambdaG" }{TEXT 212 18 " x), c. sinh(" }{XPPEDIT 18 0 "lambda;" " 6#%'lambdaG" }{TEXT 212 14 " x ) and cosh(" }{XPPEDIT 18 0 "lambda;" " 6#%'lambdaG" }{TEXT 212 4 " x)," }}{PARA 201 "" 0 "" {TEXT 212 7 "b. s in(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 12 " x) and cos (" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 21 " x ), \+ d. sinh(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 15 " x ) a nd sinh( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 2 " (" } {XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT 212 7 " - x))." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 " " {TEXT 213 11 "Question 2." }{TEXT 212 9 " Suppose " }{XPPEDIT 18 0 " lambda;" "6#%'lambdaG" }{TEXT 212 76 " > 0. Which of these is a pair o f linearly independent solutions for Y '' + " }{XPPEDIT 18 0 "lambda^2 ;" "6#*$%'lambdaG\"\"#" }{TEXT 212 14 " Y = 0 on [0, " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT 212 2 "]?" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }} {PARA 201 "" 0 "" {TEXT 212 7 "a. exp(" }{XPPEDIT 18 0 "lambda;" "6#%' lambdaG" }{TEXT 212 14 " x) and exp( -" }{XPPEDIT 18 0 "lambda;" "6#%' lambdaG" }{TEXT 212 18 " x), c. sinh(" }{XPPEDIT 18 0 "lambda;" " 6#%'lambdaG" }{TEXT 212 14 " x ) and cosh(" }{XPPEDIT 18 0 "lambda;" " 6#%'lambdaG" }{TEXT 212 4 " x)," }}{PARA 201 "" 0 "" {TEXT 212 7 "b. s in(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 12 " x) and cos (" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 21 " x ), \+ d. sinh(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 15 " x ) a nd sinh( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 2 " (" } {XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT 212 7 " - x))." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 212 202 "Remark: If you hav e forgotten enough calculus that you do not know how to find the answe r to the above two questions, you need to do two things: review and us e Maple in some manner such as the following" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 20 "y:=x->exp(lambda*x);" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 29 "diff(y(x),x,x)-lambda^2*y(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT -1 0 "" }}}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 213 11 "Question 3." }{TEXT 212 9 " Suppose " } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 54 " > 0. Which of th ese is a bounded solution for Y '' - " }{XPPEDIT 18 0 "lambda^2;" "6#* $%'lambdaG\"\"#" }{TEXT 212 14 " Y = 0 on [0, " }{XPPEDIT 18 0 "infini ty;" "6#%)infinityG" }{TEXT 212 2 ")?" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 212 7 "a. exp(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 15 " x) b. exp(-" }{XPPEDIT 18 0 "lambda; " "6#%'lambdaG" }{TEXT 212 16 " x) c. sinh(" }{XPPEDIT 18 0 "lambd a;" "6#%'lambdaG" }{TEXT 212 17 " x) d. cosh(" }{XPPEDIT 18 0 "la mbda;" "6#%'lambdaG" }{TEXT 212 3 " x)" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 212 94 "There are two issues here: whic h is a solution and which is bounded on the specified interval." }} {PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }} {PARA 201 "" 0 "" {TEXT 213 11 "Question 4." }{TEXT 212 90 " Which of \+ these is a bounded solution on the interval [0, 5] for the differentia l equation" }}{PARA 201 "" 0 "" {TEXT 212 10 " " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT 212 33 " R ''(r) + r R '(r) - 9 R( r) = 0?" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 212 21 "a. exp(3 r) b. " }{XPPEDIT 18 0 "r^3;" "6#*$%\"rG\"\"$" }{TEXT 212 49 " c. sin(3 r) d. exp(-3 r) e. 1/" } {XPPEDIT 18 0 "r^3;" "6#*$%\"rG\"\"$" }{TEXT 212 19 " f. cosh(3 \+ r)" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 213 11 "Question 5." }{TEXT 212 14 " If u(x,y) = " }{XPPEDIT 18 0 "sum(a[ p]*sin(p*x)*sinh(p*y));" "6#-%$sumG6#*(&%\"aG6#%\"pG\"\"\"-%$sinG6#*&F *F+%\"xGF+F+-%%sinhG6#*&F*F+%\"yGF+F+" }{TEXT 212 3 " + " }{XPPEDIT 18 0 "sum(b[p]*sin(p*x)*sinh(p*(Pi-y)));" "6#-%$sumG6#*(&%\"bG6#%\"pG \"\"\"-%$sinG6#*&F*F+%\"xGF+F+-%%sinhG6#*&F*F+,&%#PiGF+%\"yG!\"\"F+F+ " }{TEXT 212 6 " and " }}{PARA 201 "" 0 "" {TEXT 212 43 " \+ u(x,0) = 0, u(x," }{XPPEDIT 18 0 "pi;" "6#%#piG" } {TEXT 212 13 ") = sin(2 x) " }}{PARA 201 "" 0 "" {TEXT 212 13 "what ar e the " }{XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG" }{TEXT 212 13 " 's \+ and " }{XPPEDIT 18 0 "b[p];" "6#&%\"bG6#%\"pG" }{TEXT 212 4 " 's?" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 54 "u:=(x,y)->sum(a[p]*sin(p *x)*sinh(p*y),p=1..infinity)+ " }{MPLTEXT 1 0 62 "\n sum(b [p]*sin(p*x)*sinh(p*(Pi-y)),p=1..infinity);" }}}{PARA 201 "" 0 "" {TEXT 212 34 "We have two pieces of information." }}{EXCHG {PARA 203 " > " 0 "" {MPLTEXT 1 0 9 "0=u(x,0);" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 17 "sin(2*x)=u(x,Pi);" }}}{PARA 201 "" 0 "" {TEXT 212 40 "The first of these implies that all the " }{XPPEDIT 18 0 "b[p];" "6#& %\"bG6#%\"pG" }{TEXT 212 49 "'s are zero, and the second implies that \+ all the " }{XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG" }{TEXT 212 52 "'s i s zero except the second one, and it is 1/sinh(" }{XPPEDIT 18 0 "2*pi ;" "6#*&\"\"#\"\"\"%#piGF%" }{TEXT 212 30 "). Thus here is a graph for u." }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 97 "plot3d(1/sinh(2*Pi) *sin(2*x)*sinh(2*y), x=0..Pi,y=0..Pi,axes=NORMAL,\n orientation= [40,75]);" }}}{EXCHG {PARA 203 "" 0 "" {TEXT -1 0 "" }}}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 213 11 "Question 6." } {TEXT 212 9 " If u(r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 212 4 ") = " }{XPPEDIT 18 0 "sum(a[p]*sin(p*theta)*r^p,p);" "6#-%$sumG 6$*(&%\"aG6#%\"pG\"\"\"-%$sinG6#*&F*F+%&thetaGF+F+)%\"rGF*F+F*" } {TEXT 212 3 " + " }{XPPEDIT 18 0 "sum(b[p]*cos(p*theta)*r^p,p);" "6#-% $sumG6$*(&%\"bG6#%\"pG\"\"\"-%$cosG6#*&F*F+%&thetaGF+F+)%\"rGF*F+F*" } {TEXT 212 6 " and " }}{PARA 201 "" 0 "" {TEXT 212 10 " u(1, " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 212 16 ") = 1 + 3 cos(3 " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 212 13 ") + 5 sin( 2 " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 212 2 ") " }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 212 18 "then what is u(r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 212 22 "), u(0,0), and \+ u(1/2, " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT 212 4 "/4)?" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 57 "u:=(r,theta)->sum(a[p]*sin(p*theta)*r^p,p=1..infinity) + " } {MPLTEXT 1 0 60 "\n sum(b[p]*cos(p*theta)*r^p,p=0..in finity);" }}}{PARA 201 "" 0 "" {TEXT 212 18 "Our information is" }} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 43 "1+3*cos(3*theta)+5*sin(2*t heta)=u(1,theta);" }}}{PARA 201 "" 0 "" {TEXT 212 22 "This implies tha t all " }{XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG" }{TEXT 212 7 "'s and \+ " }{XPPEDIT 18 0 "b[p];" "6#&%\"bG6#%\"pG" }{TEXT 212 19 "'s are zero \+ except " }{XPPEDIT 18 0 "b[0];" "6#&%\"bG6#\"\"!" }{TEXT 212 6 " = 1, \+ " }{XPPEDIT 18 0 "b[3];" "6#&%\"bG6#\"\"$" }{TEXT 212 10 " = 3, and " }{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT 212 5 " = 5." }} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 54 "u:=(r,theta)->5*sin(2*thet a)*r^2+1+3*cos(3*theta)*r^3;" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 7 "u(0,0);" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 12 "u(1/2,P i/4);" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 49 "plot3d([r,theta, u(r,theta)],r=0..1,theta=-Pi..Pi," }{MPLTEXT 1 0 65 "\n co ords=cylindrical,axes=normal,orientation=[20,40]," }{MPLTEXT 1 0 30 " \n numpoints=2000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT -1 0 "" }}}{PARA 201 "" 0 " " {TEXT 212 5 " " }}{PARA 201 "" 0 "" {TEXT 213 11 "Question 7." } {TEXT 212 73 " What are all the eigenvalues of the selfadjoint, Sturm- Liouville Problem" }}{PARA 201 "" 0 "" {TEXT 212 17 " y '' = \+ " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT 212 25 " y, with y(0) = y(1) = 0?" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 212 57 "We break the problem into two cases. First, suppose that " } {XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT 212 41 " is positive. To remind u s of this, take " }{XPPEDIT 18 0 "mu = lambda^2;" "6#/%#muG*$%'lambdaG \"\"#" }{TEXT 212 25 " . Thus, we seek numbers " }{XPPEDIT 18 0 "lambd a;" "6#%'lambdaG" }{TEXT 212 10 " such that" }}{PARA 201 "" 0 "" {TEXT 212 17 " y '' = " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lam bdaG\"\"#" }{TEXT 212 24 " y with y(0) = y(1) = 0." }}{PARA 201 "" 0 " " {TEXT 212 186 "The differential equation is a second order constant \+ coefficient equation. We know how to solve it. Solutions are of the fo rm exp(r x). In this case, the general solutions is of the form" }} {PARA 201 "" 0 "" {TEXT 212 29 " y(x) = A exp( " } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 13 " x) + B exp( " } {XPPEDIT 18 0 "-lambda;" "6#,$%'lambdaG!\"\"" }{TEXT 212 4 " x)." }} {PARA 201 "" 0 "" {TEXT 212 83 "To ask that y(0) = 0 asks that 0 = A + B. To ask that y(1) = 0 asks that 0 = A exp(" }{XPPEDIT 18 0 "lambda; " "6#%'lambdaG" }{TEXT 212 10 ") + B/exp(" }{XPPEDIT 18 0 "lambda;" "6 #%'lambdaG" }{TEXT 212 100 ") . The only solution is A = 0 = B. This i s the trivial solutions and will not be of interest to us." }}{PARA 201 "" 0 "" {TEXT 212 29 " The second case is that " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT 212 41 " is negative. To remind us of this, t ake " }{XPPEDIT 18 0 "mu = -lambda^2;" "6#/%#muG,$*$%'lambdaG\"\"#!\" \"" }{TEXT 212 24 ". Thus, we seek numbers " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 11 " such that" }}{PARA 201 "" 0 "" {TEXT 212 17 " y '' = " }{XPPEDIT 18 0 "-lambda^2;" "6#,$*$%'lambda G\"\"#!\"\"" }{TEXT 212 24 " y with y(0) = y(1) = 0." }}{PARA 201 "" 0 "" {TEXT 212 79 "We also know how to solve this differential equatio n. Solutions are of the form" }}{PARA 201 "" 0 "" {TEXT 212 28 " \+ y(x) = A sin(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 12 " x) + B cos(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 4 " x)." }}{PARA 201 "" 0 "" {TEXT 212 84 "To ask that y(0) = 0 as ks that 0 = B. To also ask that y(1) = 0 asks that 0 = A sin(" } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 82 "). We don't want \+ to have the trivial solutions again, so it must be that 0 = sin(" } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 212 81 ") . But we know \+ all the places where the sine function is zero. It must be that " } {XPPEDIT 18 0 "lambda = n*Pi;" "6#/%'lambdaG*&%\"nG\"\"\"%#PiGF'" } {TEXT 212 6 ", and " }{XPPEDIT 18 0 "mu = -n^2*Pi^2;" "6#/%#muG,$*&%\" nG\"\"#%#PiGF(!\"\"" }{TEXT 212 1 "." }}{PARA 201 "" 0 "" {TEXT 212 55 "The answer to the question is that the eigenvalues are " } {XPPEDIT 18 0 "-n^2*Pi^2;" "6#,$*&%\"nG\"\"#%#PiGF&!\"\"" }{TEXT 212 32 " and the eigenfuntions are sin(" }{XPPEDIT 18 0 "n*Pi*x;" "6#*(% \"nG\"\"\"%#PiGF%%\"xGF%" }{TEXT 212 2 ")." }}{PARA 201 "" 0 "" {TEXT 212 29 "Somehow, this is no surprise." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 213 11 "Assignment:" }{TEXT 212 69 " Find all the eigenvalues of the selfadjoint, Sturm-Liouville Problem" }} {PARA 201 "" 0 "" {TEXT 212 17 " y '' = " }{XPPEDIT 18 0 "mu; " "6#%#muG" }{TEXT 212 29 " y, with y '(0) = y '(1) = 0?" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 " " 0 "" {TEXT -1 0 "" }}{PARA 204 "" 0 "" {TEXT -1 0 "" }}}{MARK "1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }