{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Subitem" 14 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 3 12 0 0 0 0 270 0 }} {SECT 0 {EXCHG {PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 48 "Unit 28 -- Applica tion: Unforced Harmonic Motion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 264 "" 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.m ath.sc.edu/~meade/" "" }}{PARA 257 "" 0 "" {URLLINK 17 "Industrial Mat hematics Institute" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 258 " " 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.e du/" "" }}{PARA 259 "" 0 "" {URLLINK 17 "University of South Carolina " 4 "http://www.sc.edu/" "" }}{PARA 260 "" 0 "" {TEXT -1 19 "Columbia, SC 29208\n" }}{PARA 262 "" 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "ht tp://www.math.sc.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }} {PARA 263 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 38 "Copyright \251 \+ 2001 by Douglas B. Meade" }}{PARA 265 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 67 "--------------------------------------------------------- ----------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Outline of Unit 28" }}{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "28.A" 1 "" "28.A" } {TEXT -1 25 " Unforced Harmonic Motion" }}{PARA 270 "" 0 "" {HYPERLNK 17 "28.A-1" 1 "" "28.A-1" }{TEXT -1 30 " Case 1: Underdamped Motion ( \+ " }{XPPEDIT 18 0 "b^2 - 4*a*c" "6#,&*$%\"bG\"\"#\"\"\"*(\"\"%F'%\"aGF' %\"cGF'!\"\"" }{TEXT -1 6 " < 0 )" }}{PARA 270 "" 0 "" {HYPERLNK 17 "2 8.A-2" 1 "" "28.A-2" }{TEXT -1 36 " Case 2: Critically Damped Motion ( " }{XPPEDIT 18 0 "b^2 - 4*a*c" "6#,&*$%\"bG\"\"#\"\"\"*(\"\"%F'%\"aGF '%\"cGF'!\"\"" }{TEXT -1 6 " = 0 )" }}{PARA 270 "" 0 "" {HYPERLNK 17 " 28.A-3" 1 "" "28.A-3" }{TEXT -1 29 " Case 3: Overdamped Motion ( " } {XPPEDIT 18 0 "b^2 - 4*a*c" "6#,&*$%\"bG\"\"#\"\"\"*(\"\"%F'%\"aGF'%\" cGF'!\"\"" }{TEXT -1 6 " > 0 )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initialization" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg \+ ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "28.A" {TEXT -1 29 "28.A Unforced Harmonic Motion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "The general form of the ODE for unforced, or free, harmonic motion is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "ode1 := a * diff( x(t), t$2 \+ ) + b * diff( x(t), t ) + c * x(t) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "For a spring- mass system, " }{XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"tG" }{TEXT -1 30 " \+ represents the displacement, " }{XPPEDIT 18 0 "Diff( x, t )" "6#-%%Dif fG6$%\"xG%\"tG" }{TEXT -1 18 " is the velocity, " }{XPPEDIT 18 0 "a" " 6#%\"aG" }{TEXT -1 14 " is the mass, " }{XPPEDIT 18 0 "b" "6#%\"bG" } {TEXT -1 30 " is the damping constant, and " }{XPPEDIT 18 0 "c" "6#%\" cG" }{TEXT -1 132 " is the spring constant (assuming a linear spring t hat obeys Hooke's Law). The same model arises for an RLC electrical ci rcuit (see " }{HYPERLNK 17 "Unit 21" 1 "unit21.mws" "" }{TEXT -1 16 ") . In this case " }{XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"tG" }{TEXT -1 24 " represents the charge, " }{XPPEDIT 18 0 "Diff( x, t )" "6#-%%DiffG6$ %\"xG%\"tG" }{TEXT -1 17 " is the current, " }{XPPEDIT 18 0 "a" "6#%\" aG" }{TEXT -1 20 " is the inductance, " }{XPPEDIT 18 0 "b" "6#%\"bG" } {TEXT -1 24 " is the resistance, and " }{XPPEDIT 18 0 "1/c" "6#*&\"\" \"F$%\"cG!\"\"" }{TEXT -1 21 " is the capacitance (" }{XPPEDIT 18 0 "c " "6#%\"cG" }{TEXT -1 59 " is the elastance). Note that in either case , the constant " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 17 " is positi ve and " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 18 " are non-negative." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "The purpose of this wo rksheet is to present a thorough analysis of the homogeneous problem. \+ The methods introduced in " }{HYPERLNK 17 "Unit 19" 1 "unit19.mws" "" }{TEXT -1 100 "; can be applied to obtain the solution to correspondin g nonhomogeneous problems; the discussion in " }{HYPERLNK 17 "Unit 35 " 1 "unit35" "" }{TEXT -1 53 " focuses on periodic forcing functions a nd resonance." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "The solution procedure for a second-order linear ODE wit h constant coefficients calls for a search for exponential solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "q1 := factor( eval( ode1, x(t)=exp(lambda*t) ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The characteristic equation is" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "chareqn := q1 / exp( la mbda*t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "and the corresponding characteristic valu es are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "charvals := solve( chareqn, \{lambda\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "28.A-1" {TEXT -1 36 "28.A-1 Case 1: Underdamped Motion ( " }{XPPEDIT 18 0 "b^2 - 4*a*c" "6#,&*$%\"bG\"\"#\"\"\"*(\"\"%F'%\"aGF'%\"cGF'!\"\"" }{TEXT -1 6 " < 0 )" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "b^2-4*a*c" "6#,&*$%\"bG\"\"#\"\"\"*(\"\"%F'%\"aGF'%\"cGF'!\"\"" }{TEXT -1 77 " < 0, the characteristic values are complex conjugates t hat can be written as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "lambda[1] := alpha + beta*I;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "lambda[2] := alpha - beta*I; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "alpha = b/2" "6#/%&alphaG*&% \"bG\"\"\"\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta = sqrt( 4*a*c - b^2 )" "6#/%%betaG-%%sqrtG6#,&*(\"\"%\"\"\"%\"aGF+%\"cGF+F+*$ %\"bG\"\"#!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "The general solution i s a linear combination of exponentially damped trigonometric functions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "X[1] := exp( -alpha * t ) * cos( beta * t );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "X[2] := exp( -alpha * t ) * sin( be ta * t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "X[h] := c[1] * X[1] + c[2] * X[2]; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "or, equivalently," }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "X[h2] := A * exp( -alpha*t ) * sin( beta*t + phi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "A = s qrt( c[1]^2 + c[2]^2 )" "6#/%\"AG-%%sqrtG6#,&*$&%\"cG6#\"\"\"\"\"#F-*$ &F+6#F.F.F-" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(phi)=c[1]/A" "6#/-%$ sinG6#%$phiG*&&%\"cG6#\"\"\"F,%\"AG!\"\"" }{TEXT -1 6 ", and " } {XPPEDIT 18 0 "cos(phi)=c[2]/A" "6#/-%$cosG6#%$phiG*&&%\"cG6#\"\"#\"\" \"%\"AG!\"\"" }{TEXT -1 16 ". In this form, " }{XPPEDIT 18 0 "A*exp(-a lpha*t)" "6#*&%\"AG\"\"\"-%$expG6#,$*&%&alphaGF%%\"tGF%!\"\"F%" } {TEXT -1 26 " is the damped amplitude, " }{XPPEDIT 18 0 "beta/2/Pi" "6 #*(%%betaG\"\"\"\"\"#!\"\"%#PiGF'" }{TEXT -1 25 " is the quasi-frequen cy, " }{XPPEDIT 18 0 "2*Pi/beta" "6#*(\"\"#\"\"\"%#PiGF%%%betaG!\"\"" }{TEXT -1 26 " is the quasi-period, and " }{XPPEDIT 18 0 "phi" "6#%$ph iG" }{TEXT -1 20 " is the phase angle." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 84 "The decaying exponential term forces \+ all solutions to decay to the zero function as " }{XPPEDIT 18 0 "t" "6 #%\"tG" }{TEXT -1 11 " increases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "2 8.A-2" {TEXT -1 42 "28.A-2 Case 2: Critically Damped Motion ( " } {XPPEDIT 18 0 "b^2-4*a*c" "6#,&*$%\"bG\"\"#\"\"\"*(\"\"%F'%\"aGF'%\"cG F'!\"\"" }{TEXT -1 6 " = 0 )" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Whe n " }{XPPEDIT 18 0 "b^2-4*a*c" "6#,&*$%\"bG\"\"#\"\"\"*(\"\"%F'%\"aGF' %\"cGF'!\"\"" }{TEXT -1 59 " = 0 there is a double root to the charact eristic equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "lambda[1] := -b / 2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "lambda[2] := -b / 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Two linearly \+ independent solutions to the ODE in this case are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "X[1] := exp ( lambda[1]*t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "X[2] := t * X[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "and the general solution is" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "X[h] := c[1]*X[1] + c[2]*X[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Subcase 1: " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 4 " > 0" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 236 "So long as b > 0 the common exponential factor overwhelms the \+ linear growth and forces all solutions to zero. The decay to zero is n ot necessarily monotonic. The general solution to the critically dampe d equation has a critical point at" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dX[h] := factor( diff( X[h] , t ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "t_crit := solve ( dX[h]=0, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "which is positive if and only if \+ " }{XPPEDIT 18 0 "c[2] <> 0" "6#0&%\"cG6#\"\"#\"\"!" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "c[1]/c[2]" "6#*&&%\"cG6#\"\"\"F'&F%6#\"\"#!\"\"" } {TEXT -1 3 " < " }{XPPEDIT 18 0 "2/b" "6#*&\"\"#\"\"\"%\"bG!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Subcas e 2: " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 4 " = 0" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Note that if " }{XPPEDIT 18 0 "b" "6#%\"bG" } {TEXT -1 11 " = 0, then " }{XPPEDIT 18 0 "4*a*c" "6#*(\"\"%\"\"\"%\"aG F%%\"cGF%" }{TEXT -1 9 " = 0 and " }{XPPEDIT 18 0 "a" "6#%\"aG" } {TEXT -1 16 " > 0 imply that " }{XPPEDIT 18 0 "c=0" "6#/%\"cG\"\"!" } {TEXT -1 36 ". In this case the ODE simplifies to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eval( ode1, [a=1,b=0,c=0] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "a fundamenta l set of solutions is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eval( X[1], b=0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eval( X[2], b=0 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "and the gener al solution is a linear function. Hence, solutions do not tend to zero as " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 122 " increases. In fact, except in special cases where the solution is a constant function, th e solutions become unbounded as " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 11 " increases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 4 "" 0 "28.A-3" {TEXT -1 35 "28.A-3 Case 3: Overdamped Motion ( " }{XPPEDIT 18 0 "b^2 - 4*a* c" "6#,&*$%\"bG\"\"#\"\"\"*(\"\"%F'%\"aGF'%\"cGF'!\"\"" }{TEXT -1 6 " \+ > 0 )" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The third case is the mos t straightforward. If " }{XPPEDIT 18 0 "b^2-4*a*c" "6#,&*$%\"bG\"\"#\" \"\"*(\"\"%F'%\"aGF'%\"cGF'!\"\"" }{TEXT -1 61 " > 0, then there are t wo distinct real characteristic values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "lambda[1] := eval( lam bda, charvals[1] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "lambda[2] := eval( lambda, charvals[2] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "that produce a fundament al set of solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "X[1] := exp( lambda[1]*t );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "X[2] := exp( lambda[2]*t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "a nd general solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "X[h] := c[1]*X[1] + c[2]*X[2];" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Ob serve that, because 0 < " }{XPPEDIT 18 0 "b^2-4*a*c" "6#,&*$%\"bG\"\"# \"\"\"*(\"\"%F'%\"aGF'%\"cGF'!\"\"" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "b^2" "6#*$%\"bG\"\"#" }{TEXT -1 52 ", both characteristic values are \+ negative. In fact, " }{XPPEDIT 18 0 "lambda[2]" "6#&%'lambdaG6#\"\"#" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "lambda[1]" "6#&%'lambdaG6#\"\"\"" } {TEXT -1 174 " < 0. All solutions to an overdamped equation ultimately decay to zero as t increases. But, just as with the critically damped case, the solution can have one critical point:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dX[h] := di ff( X[h], t ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "tcrit := solve( d X[h] = 0, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "To determine when the critical poi nt occurs with " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 74 " > 0, obse rve that the location of the critical point can be expressed as " } {XPPEDIT 18 0 "t[crit]" "6#&%\"tG6#%%critG" }{TEXT -1 5 " = - " } {XPPEDIT 18 0 "a / sqrt( b^2-4*a*c)" "6#*&%\"aG\"\"\"-%%sqrtG6#,&*$%\" bG\"\"#F%*(\"\"%F%F$F%%\"cGF%!\"\"F0" }{TEXT -1 6 " ln( " }{XPPEDIT 18 0 "c[1]/c[2]" "6#*&&%\"cG6#\"\"\"F'&F%6#\"\"#!\"\"" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "lambda[1]/lambda[2]" "6#*&&%'lambdaG6#\"\"\"F'&F%6# \"\"#!\"\"" }{TEXT -1 45 " ). This time is positive precisely when 0 < " }{XPPEDIT 18 0 "c[1]/c[2]" "6#*&&%\"cG6#\"\"\"F'&F%6#\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda[1]/lambda[2]" "6#*&&%'lambdaG6# \"\"\"F'&F%6#\"\"#!\"\"" }{TEXT -1 5 " < 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "To conclude, note that except for the tri vial case with " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 209 " = 0, all solutions to the \+ ODE for simple harmonic motion are transient solutions. This means tha t the steady-state behavior of a solution for a forced harmonic motion problem depends on the forcing function, " }{TEXT 256 4 "i.e." } {TEXT -1 57 ", on a particular solution to the nonhomogeneous problem. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " } {HYPERLNK 17 "ODE Powertool Table of Contents" 1 "unit00.mws" "" } {TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }