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{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.-%$cosGF)F.F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "I
f we impose particular initial conditions, we can get a particular sol
ution which we can then plot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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#%\"tG,&-%$sinGF(\"\"\"-%$cosGF(F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "plot(rhs(sol11),t=0..4*Pi, numpoints=2000);" }}{PARA 
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218 "Notice that the curve starts at the given initial condition and h
as maximum value approximately 1.42.   It should be easy for you to co
mpute a more precise value for this maximum value or AMPLITUDE of the \+
osciallation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 148 "Now, we are goint to look at the forced oscillator with \+
a sinusoidal forcing term.  We will take the same initial conditions f
or ease in comparison." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
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