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:vn]ZXSF97$$\"3aD&3O;=jp\"Fg[m$\"3?#o$f3uXeIF97$$\"33>p<)3qpp\"Fg[m$\"
3K;&**yvW^.#F97$$\"3k7`u7?i(p\"Fg[m$\"3mT\"3/PA9!**F37$$\"3[1PJPRF)p\"
Fg[m$!3+QvJ+AodhF[p7$$\"3U6Fzn)o*)p\"Fg[m$!3s\"3EDR%4t6F97$$\"31;S'H+3-P'F97$$
\"3d'))puQ^@r\"Fg[m$!3C)Hc5qG+)\\F97$$\"3`Qfro1`8'*\\*4\\r\"Fg[m$!3Z[]m%eb8h\"F97$$\"3)=/37B*G;Y\\2F/s\"Fg[m$\"3;\"f\"Gg%F97$$\"3]\"*o9xC!>s\"
Fg[m$\"3EPMkm4eZbF97$$\"3C&eZ$zyPB$zW!e
:Ts\"Fg[m$\"3+8d\\!3@u8'F97$$\"3py#[:G`[s\"Fg[m$\"3G[?\\ax#*4hF97$$\"3
TD'[E)4fDP;U+u\"Fg[m$!3%*o+/hgG.GF97$$\"3waZx?hkU.g " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The derivative of the \+
solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`y '
`(t) = `g '`(t)+2;" "6#/-%$y~'G6#%\"tG,&-%$g~'G6#F'\"\"\"\"\"#F," }
{XPPEDIT 18 0 "Sum((-1)^k*u[2*k-1](t)*`g '`(t-2*k+1),k = 1 .. infinity
);" "6#-%$SumG6$*(),$\"\"\"!\"\"%\"kGF)-&%\"uG6#,&*&\"\"#F)F+F)F)F)F*6
#%\"tGF)-%$g~'G6#,(F4F)*&F2F)F+F)F*F)F)F)/F+;F)%)infinityG" }{TEXT -1
2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT
-1 1 " " }{XPPEDIT 18 0 "`y '`(t) = `g '`(t)+2;" "6#/-%$y~'G6#%\"tG,&-
%$g~'G6#F'\"\"\"\"\"#F," }{XPPEDIT 18 0 "Sum((-1)^i*`g '`(t-2*i+1),i =
1 .. floor((t+1)/2));" "6#-%$SumG6$*&),$\"\"\"!\"\"%\"iGF)-%$g~'G6#,(
%\"tGF)*&\"\"#F)F+F)F*F)F)F)/F+;F)-%&floorG6#*&,&F0F)F)F)F)F2F*" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "
`g '`(t) = 1/4-cos*2*t/4;" "6#/-%$g~'G6#%\"tG,&*&\"\"\"F*\"\"%!\"\"F**
*%$cosGF*\"\"#F*F'F*F+F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT
-1 67 "We can draw the associated phase curve for the solution as foll
ows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 265 "f := t -> t/4-sin(2*t)/8:\nh := t -> 'f(t)'+2*Sum((-
1)^k*'f(t-2*k+1)',k=1..floor((t+1)/2)):\ndf := t -> 1/4-cos(2*t)/4:\nd
h := t -> 'df(t)'+2*Sum((-1)^k*'df(t-2*k+1)',k=1..floor((t+1)/2)):\npl
ot([h(t),dh(t),t=0..176.54],color=COLOR(RGB,.5,0,1),labels=[`y(t)`,`y'
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RcF0$!3HbvX8_a< " 0 ""
{MPLTEXT 1 0 318 "f := t -> (-t+t*(-1)^floor((t+1)/2)-2*(-1)^floor((t+
1)/2)*floor((t+1)/2))*Heaviside(t-1)+t:\nde := diff(y(t),t$2)+4*y(t)=f
(t);\nic := y(0)=0,D(y)(0)=0;\nsol := dsolve(\{de,ic\},y(t),type=numer
ic,abserr=1e-11,\n relerr=1e-11,output=listpr
ocedure):\nhn := subs(sol,y(t));\ndhn := subs(sol,diff(y(t),t));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"tG-%\"$G
6$F-\"\"#\"\"\"*&\"\"%F2F*F2F2,&*&,(F-!\"\"*&F-F2)F8-%&floorG6#,&*&F1F
8F-F2F2#F2F1F2F2F2*(F1F2F:F2F;F2F8F2-%*HeavisideG6#,&F-F2F2F8F2F2F-F2
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#
F(F)F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 171 "g := t -> t/4-sin(2*t)/8:\nh := t -> 'g(t)'+2*Sum((-
1)^k*'g(t-2*k+1)',k=1..floor((t+1)/2)):\nplot([h(t),'hn'(t)],t=0..44.5
,thickness=[1,2],labels=[`t`,`y(t)`],ytickmarks=3);" }}{PARA 13 "" 1 "
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