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Unformatted text preview: {VERSION 5 0 "SUN SPARC SOLARIS" "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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" 1 0 1 {CSTYLE "" 1 1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 1 1 1 0 0 0 0 0 0 1 0 }{PSTYLE "Text Output" 1 2 1 {CSTYLE "" 1 1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 1 }1 0 0 1 1 1 0 0 0 0 0 0 1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" 1 1 " " 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 1 }0 0 0 1 1 1 0 0 0 0 0 0 1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" 1 1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 1 1 1 0 0 0 0 0 0 1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" 1 1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 1 1 1 0 0 0 0 0 0 1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" 1 1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 1 1 1 0 0 0 0 0 0 1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 1 0 "" }}{PARA 0 "" 0 "" {TEXT 1 56 " BEATS AND RESONANCE" }} {PARA 0 "" 0 "" {TEXT 1 0 "" }}{PARA 0 "" 0 "" {TEXT 1 756 "We have \+ looked at the Method of Undetermined Coefficients for nonhomogeneous s econd order constant coefficient ODE. In this sheet we want to look s pecifically at the problem of the Harmonic Oscillator and the Damped H armonic Oscillator and at sinusoidal (periodic) forcing terms. We exp ect that, if the forcing term is a solution of the homogeneous equatio n, which in this case means that the frequency of the forcing term is \+ the same as the natural frequency of the system, we should expect that we will find solutions that grow with t . This is the phenomenon of resonance. We will start with the situation in which the frequencies are different, and begin comparing the results as the frequency of th e forcing term approaches the natural frequency." }}{PARA 0 "" 0 "" {TEXT 1 0 "" }}{PARA 0 "" 0 "" {TEXT 1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT 1 50 "Warning, the name c hangecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1 0 "" }}{PARA 0 "" 0 "" {TEXT 1 43 "Let us start with the harmonic osc illator :" }}{PARA 0 "" 0 "" {TEXT 1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "de1:=diff(y(t),t$2) + y(t)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/,&%%diffG6$%\"yG6#%\"tG%\"$G6$F\"\"#\"\"\"F *F2\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1 0 "" }}{PARA 0 "" 0 "" {TEXT 1 44 "This simple equation has a general solution " }}{PARA 0 " " 0 "" {TEXT 1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sol1 :=dsolve(\{de1\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G<#/%\"yG6#%\"tG,&*&%$_C1G\"\"\"%$sinGF)F.F.*&%$_C2GF.%$cosGF)F.F." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 1 0 "" }}{PARA 0 "" 0 "" {TEXT 1 100 "I f we impose particular initial conditions, we can get a particular sol ution which we can then plot." }}{PARA 0 "" 0 "" {TEXT 1 0 "" }}} ution which we can then plot....
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This note was uploaded on 12/02/2009 for the course MATH 352 taught by Professor Staff during the Spring '08 term at University of Delaware.
 Spring '08
 Staff
 Math

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