Math 2650 - Linear Differential EquationsA. J. MeirCopyright (C) A. J. Meir. All rights reserved.This worksheet is for educational use only. No part of this publication may be reproduced or transmitted for profit in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system without prior written permission from the author. Not for profit distribution of the software is allowed without prior written permission, providing that the worksheet is not modified in any way and full credit to the author is acknowledged.

We consider equations of the typeNiMlIm1HNiMvLCYqKCUiZEciIiMlInhHIiIiKiQlI2R0R0YnISIiRiklI2t4R0YpKiYlIkZHRiktJSRjb3NHNiMqJiUmb21lZ2FHRiklInRHRilGKQ==.For example take NiMvJSJtRyIiIg==, NiMvJSJrRyIiKg==, and NiMvJSJGRyIiJQ== (the roots of the characteristic polynomial are NiMqJiIiJCIiIiUiSUdGJQ== and NiMsJComIiIkIiIiJSJJR0YmISIi)NiMvLCYqKCUiZEciIiMlInhHIiIiKiQlI2R0R0YnISIiRikqJiIiKkYpRihGKUYpKiYiIiVGKS0lJGNvc0c2IyomJSZvbWVnYUdGKSUidEdGKUYpwith initial conditionsNiMvLSUieEc2IyIiIUYn and NiMqJiUjZHhHIiIiJSNkdEchIiI= (0)=0.restart:with(DEtools):dsolve({D(D(x))(t)+9*x(t)=4*cos(omega*t),x(0)=0,D(x)(0)=0},x(t));

NiMvLUkieEc2IjYjSSJ0R0YmLCYqJi1JJGNvc0c2JEkqcHJvdGVjdGVkR0YuSShfc3lzbGliR0YmNiMsJEYoIiIkIiIiLCYhIipGMyokSSZvbWVnYUdGJiIiI0YzISIiIiIlKiYtRiw2IyomRjdGM0YoRjNGM0Y0RjkhIiU=

That is given a periodic function of the formNiMvLSUiZkc2IyUidEcsJiomJSJhRyIiIi0lJGNvc0c2IyomJSZvbWVnYUdGK0YnRitGK0YrKiYlImJHRistJSRzaW5HRi5GK0Yrwe can write it asNiMvLSUiZkc2IyUidEcqJiUiQUciIiItJSRjb3NHNiMsJiomJSZvbWVnYUdGKkYnRipGKiUmZGVsdGFHISIiRio=whereNiMvJSJBRy0lJXNxcnRHNiMsJiokJSJhRyIiIyIiIiokJSJiR0YrRiw=andNiMvJSZkZWx0YUctJSdhcmN0YW5HNiMqJiUiYkciIiIlImFHISIi(note this last equation has two solutions which differ by NiMlI1BpRw==, you must chose the correct solution).Also recall the two trigonometric identitiesNiMvLCYtJSRjb3NHNiMlJmFscGhhRyIiIi1GJjYjJSViZXRhRyEiIiwkKigiIiNGKS0lJHNpbkc2IyomLCZGKEYpRixGKUYpRjBGLUYpLUYyNiMqJiwmRihGKUYsRi1GKUYwRi1GKUYtandNiMvLCYtJSRzaW5HNiMlJmFscGhhRyIiIi1GJjYjJSViZXRhRyEiIiooIiIjRiktJSRjb3NHNiMqJiwmRihGKUYsRilGKUYvRi1GKS1GJjYjKiYsJkYoRilGLEYtRilGL0YtRik=using these identities we can often rewrite periodic functions to better see the phenomenon of beats.

sol(t):=-8/(-9+omega^2)*sin((3-omega)*t/2)*sin((3+omega)*t/2);

NiM+LUkkc29sRzYiNiNJInRHRiYsJCooLCYhIioiIiIqJEkmb21lZ2FHRiYiIiNGLSEiIi1JJHNpbkc2JEkqcHJvdGVjdGVkR0Y1SShfc3lzbGliR0YmNiMsJComLCYiIiRGLUYvRjFGLUYoRi0jRi1GMEYtLUYzNiMsJComLCZGO0YtRi9GLUYtRihGLUY8Ri0hIik=

env(t):=-8/(-9+omega^2)*sin((3-omega)*t/2);

NiM+LUkkZW52RzYiNiNJInRHRiYsJComLCYhIioiIiIqJEkmb21lZ2FHRiYiIiNGLSEiIi1JJHNpbkc2JEkqcHJvdGVjdGVkR0Y1SShfc3lzbGliR0YmNiMsJComLCYiIiRGLUYvRjFGLUYoRi0jRi1GMEYtISIp

omega:=1:plot([sol(t),env(t),-env(t)],t=0..20,color=[black,red,red]);

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omega:=2:plot([sol(t),env(t),-env(t)],t=0..20,color=[black,red,red]);

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omega:=5/2:plot([sol(t),env(t),-env(t)],t=0..20,color=[black,red,red]);

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omega:=1.75*sqrt(2):plot([sol(t),env(t),-env(t)],t=0..20,color=[black,red,red]);

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omega:=29/10:plot([sol(t),env(t),-env(t)],t=0..50,color=[black,red,red]);

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For NiMvJSZvbWVnYUciIiQ= we getdsolve({D(D(x))(t)+9*x(t)=4*cos(3*t),x(0)=0,D(x)(0)=0},x(t));

NiMvLUkieEc2IjYjSSJ0R0YmLCQqJi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YuSShfc3lzbGliR0YmNiMsJEYoIiIkIiIiRihGMyMiIiNGMg==

sol(t):=2/3*sin(3*t)*t;

NiM+LUkkc29sRzYiNiNJInRHRiYsJComLUkkc2luRzYkSSpwcm90ZWN0ZWRHRi5JKF9zeXNsaWJHRiY2IywkRigiIiQiIiJGKEYzIyIiI0Yy

env(t):=2/3*t;

NiM+LUkkZW52RzYiNiNJInRHRiYsJEYoIyIiIyIiJA==

plot([sol(t),env(t),-env(t)],t=0..25,color=[black,red,red]);

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We consider equations of the typeNiMlIm1HNiMsJiooJSJkRyIiIyUieEciIiIqJCUjZHRHRiYhIiJGKCUiY0dGKA==NiMvLCYqJiUjZHhHIiIiJSNkdEchIiJGJyUja3hHRicqJiUiRkdGJy0lJGNvc0c2IyomJSZvbWVnYUdGJyUidEdGJ0YnFor example assume that NiMvJSJtRyIiIg==, NiMvJSJrRyIiIg==, NiMvJSJjRyomIiIiRiYiIikhIiI=, and NiMvJSJGRyIiJA== (the roots of the characterisatic polynomial are NiMqJiwmIiIiISIiKiYlIklHRiUtJSVzcXJ0RzYjIiRiI0YlRiVGJS1GKjYjIiRjI0Ym and NiMqJiwmIiIiISIiKiYlIklHRiUtJSVzcXJ0RzYjIiRiI0YlRiZGJS1GKjYjIiRjI0Ym).We now in addition assume that NiMvJSZvbWVnYUckIiIkISIi and solve the initial value problem with initial conditionsNiMlInhH(0)=0 NiMqJiUjZHhHIiIiJSNkdEchIiI=(0)=0restart:with(DEtools):with(plots):

Warning, the name changecoords has been redefined

de:=diff(x(t),t,t)+1/8*diff(x(t),t)+x(t)=3*cos(omega*t);

NiM+SSNkZUc2Ii8sKC1JJWRpZmZHSSpwcm90ZWN0ZWRHRio2JC1JInhHRiU2I0kidEdGJS1JIiRHRio2JEYvIiIjIiIiLUYpNiRGLEYvI0Y0IiIpRixGNCwkLUkkY29zRzYkRipJKF9zeXNsaWJHRiU2IyomSSZvbWVnYUdGJUY0Ri9GNCIiJA==

dsolve({subs(omega=3/10,de),x(0)=0,D(x)(0)=0},x(t));

NiMvLUkieEc2IjYjSSJ0R0YmLCoqKC1JJGV4cEc2JEkqcHJvdGVjdGVkR0YuSShfc3lzbGliR0YmNiMsJEYoIyEiIiIjOyIiIi1JJHNpbkdGLTYjLCQqJiIkYiMjRjUiIiNGKEY1I0Y1RjRGNUY7RjwjISYhKVskIihkaUQjKiZGK0Y1LUkkY29zR0YtRjhGNSMhJytvViInQEY4LUY3NiMsJEYoIyIiJCIjNSMiJishPUZHLUZERkkjIicrb1ZGRw==

We now identify the steady state (the term that persists in NiMlInRH) and the transient the term that decays in NiMlInRH.sol:=rhs(%);

NiM+SSRzb2xHNiIsKiooLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIyEiIiIjOyIiIi1JJHNpbkdGKjYjLCQqJiIkYiMjRjMiIiNGL0YzI0YzRjJGM0Y5RjojISYhKVskIihkaUQjKiZGKEYzLUkkY29zR0YqRjZGMyMhJytvViInQEY4LUY1NiMsJEYvIyIiJCIjNSMiJishPUZFLUZCRkcjIicrb1ZGRQ==

transient:=select(has,sol,exp);

NiM+SSp0cmFuc2llbnRHNiIsJiooLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIyEiIiIjOyIiIi1JJHNpbkdGKjYjLCQqJiIkYiMjRjMiIiNGL0YzI0YzRjJGM0Y5RjojISYhKVskIihkaUQjKiZGKEYzLUkkY29zR0YqRjZGMyMhJytvViInQEY4

steady:=remove(has,sol,exp);

NiM+SSdzdGVhZHlHNiIsJi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliR0YlNiMsJEkidEdGJSMiIiQiIzUjIiYrIT0iJ0BGOC1JJGNvc0dGKUYsIyInK29WRjQ=

We now plot the solution, steady state and transient on the same coordinate system.plot({sol,transient,steady}, t=0..40,color=[magenta,red,blue]);

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andplot({sol,transient,steady}, t=0..80,color=[magenta,red,blue]);

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L$e*)G$Q]TF\s$"3sC"z^q7d:#F27$F^il$"3)H//Gh.r"=F27$Fbjl$"3?:A"\w-Oz"F[p7$F\[m$!3Sf]A#[nWS"F27$$"3xm;/,ypYVF\s$!3L@b&ReVFq"F27$$"3TLL3ngUnVF\s$!3d)[?o9G9#>F27$$"30+]7LV:)Q%F\s$!3])Rc&[ZJ`?F27$Fa[m$!39*H4pf\]4#F27$$"3ML$3_'3hHWF\s$!3j,O")GZ<Z?F27$$"3)****\78R.X%F\s$!35>Txca+9>F27$$"3im;H(Rn5Z%F\s$!3hFYV"[^Lq"F27$Ff[m$!3'=5vaX(4E9F27$F`\m$"3*)3#H4y?s2"F[p7$Fj\m$"3[u,.+xJ:9F27$$"3s++D@+1%o%F\s$"3)[*y;Q*p4f"F27$F_]m$"3uq3.]PT$p"F27$$"3e++v.&ffs%F\s$"3ay9Un(>,s"F27$Fd]m$"3$\69%RH*=n"F27$Fi]m$"3F8(pw+r'p8F27$F^^m$"3%y./3]4,_)F[p7$Fh^m$!3ie()3,_*\#QF[p7$Fb_m$!3Qlc^Y#3<F"F27$$"3.+]i]TP:]F\s$!3DsW')\?Ty8F27$$"3L++Dw/)y.&F\s$!3`Z(R;c(f89F27$$"3j+](=!oQg]F\s$!3^yy5N()\x8F27$Fg_m$!3S94n5&))QF"F27$$"36++vyd!z7&F\s$!35`AqNr^]*)F[p7$F\`m$!3qZxHH!obj$F[p7$F`am$"3Ql)yyg,P>'F[p7$Fdbm$"3,8H-LU'G9"F27$$"3rKL$e?%Hc`F\s$"3Yr#f!)[o-;"F27$$"3K*****\]')yP&F\s$"3Eb>"o"\/C6F27$$"3kmm;/)y%*R&F\s$"37lgQ'Q"GP5F27$Fibm$"3nlwSUJ*H0*F[p7$$"3<nmm,dDkaF\s$"3)=[f/UFBO&F[p7$F^cm$"3.=ILy0Er!*!#?7$Fccm$!3cmJr0v!**3$F[p7$Fhcm$!38MkA-pASkF[p7$$"32L$e9Yp_g&F\s$!3+%[79&RJ@xF[p7$F]dm$!3wOk,2hP!o)F[p7$$"3em;/E6SWcF\s$!3(*yC(**fR!*G*F[p7$Fbdm$!3<%fJzGDG`*F[p7$$"3Zm;/"ed`o&F\s$!36*\M"H%y<Q*F[p7$Fgdm$!326xQLhu7))F[p7$$"3YL$ek#)Q"GdF\s$!3dR)fAF]B'yF[p7$F\em$!3d8Z?O$*R$e'F[p7$Faem$!36UnL3wl9LF[p7$Ffem$"3-ZHA'QZkl$Fbbq7$F[fm$"3gpU.%3%=1OF[p7$F`fm$"3+#Ru)R$ya7'F[p7$$"3um;z%[Ed$fF\s$"33dV9yB4'*pF[p7$Fefm$"3S"p2Sg[hc(F[p7$$"3jL$3FI!)f(fF\s$"3]E>G7nQ?yF[p7$Fjfm$"3mP@`1p^cxF[p7$F_gm$"3t+!HcS3.[%F[p7$Fdgm$!3cf5V@p"pI"F[p7$Figm$!3;j`$>!*3X%QF[p7$F^hm$!3.7'>orM'HcF[p7$Fchm$!3T5ws7GRahF[p7$Fhhm$!3<84^oZn4kF[p7$F]im$!3A*>@2b&p"R'F[p7$Fbim$!3G&Q0UyR"3hF[p7$Fgim$!3Wzq;gl)\h#F[p7$F\jm$"3Q?p5NF+]AF[p7$Fajm$"3)z(fG/^'*[SF[p7$Ffjm$"3nF4Y9)Gy3&F[p7$F[[n$"3AAW3V%*HL_F[p7$F`[n$"3*["zol]Z1XF[p7$Fj[n$"3X2&QU?X<6"F[p7$Fd\n$!3B:Mg<$ehl#F[p7$Fi\n$!3h%[(Qg)zR&QF[p7$F^]n$!3YKRsTtyNVF[p7$Fc]n$!3^gixrmTcSF[p7$Fh]n$!3]kffRYJ-JF[p7$F]^n$!3\!\()ypV:'HFbbq7$Fb^n$"3oX;%4?.?T#F[p7$$"3'zmmTU=S?(F\s$"3'[)*QJ+[4K$F[p7$Fg^n$"3i3<1Dv6dNF[p7$$"3;-+]7^i#H(F\s$"3_@'eOa%R7JF[p7$F\_n$"3,'*H&>.)*p5#F[p7$F``n$!3[l6_:cofOFbbq7$Fj`n$!3qk\"*Qm2)R#F[p7$F_an$!3Mxa[tCtXGF[p7$Fdan$!3bs3l6SKK9F[p7$F^bn$"3S+s6VUVXmFbbq7$Fhbn$"3`HgEp0Uo@F[p7$Fbcn$"3/\n2DO:'>#F[p7$F\dn$"36mzripYbuFbbq-Fadn6&FcdnFddnFgdnFgdn-F&6$7cv7$F*F*7$F6$"3$3\![^Hs^FF27$F@$"3kjE0p9KE5F.7$F`fn$"3)4"\wnD/G:F.7$FE$"3MCivpE_x?F.7$Fbgn$"3%4c'3AGgWEF.7$FJ$"3[@wi,,c)>$F.7$Fdhn$"3=7MYN8%Q/%F.7$FO$"3^&4c#>S!Hi%F.7$$"3Umm"zp^?p#F.$"3$Q#\O`7'*zZF.7$Fain$"3*yQ&y%p*RP[F.7$$"39++vVMOrIF.$"3YjG2u.."z%F.7$FT$"3ANq'=<=-k%F.7$Fiin$"3ii%f$zba\VF.7$FY$"3&*oeH#)fhRRF.7$Fajn$"3uZLL[XNCMF.7$Fhn$"37]<FmZy@GF.7$Fijn$"3(e!ec'[YJ:#F.7$F]o$"3'Q\RF?%*=W"F.7$Fa[o$"3mYr+zQBErF27$Fbo$!3%HTE$)HMD+"F[p7$Fi[o$!39t52,-uoqF27$Fgo$!3x-4+i:1^8F.7$Fa\o$!3!GO!\!HWN#>F.7$F]p$!37QMsnW+4CF.7$Fbp$!3=Z*fcRi$zIF.7$Fgp$!3%4FLGss+L$F.7$F\q$!3:#[!p'=\O?$F.7$Faq$!3i!z**\KSH!GF.7$Ffq$!39+$e_]?oE#F.7$F[r$!37XisZS;X<F.7$Fc`o$!3MvU[_q:"R"F.7$F`r$!36tqVxZiS7F.7$Fjao$!37*=d4DI'H8F.7$Fer$!3IS!="yxN];F.7$F`s$!3xV9v$4gW<#F.7$Fjs$!3^j[pi@;0GF.7$F_t$!3:dA!z,sGV$F.7$Fdt$!3'os)\JCmYRF.7$$"3xmmmrhC$="F\s$!3z)**zIdvb8%F.7$Fcdo$!3j%pxDtW$eUF.7$$"3ymmmJ#o]@"F\s$!3)z%zH14`#H%F.7$$"36+++&evcA"F\s$!31'*QJ%)4n2VF.7$$"3XLLLQHGO7F\s$!3V&)H)=a<LI%F.7$Fit$!3NcS1ev@zUF.7$Fjeo$!3QL$=>C#3()RF.7$F^u$!3;e)o5sP!3MF.7$$"3xmm;*y&*GN"F\s$!39D**=k3;KIF.7$Fcu$!3v%3!QyV%Hh#F.7$$"3QLL$38*=&R"F\s$!3W([tu'G:j@F.7$Fhu$!3/&p^;1nip"F.7$$"3)*****\sC[P9F\s$!3ozAI_4!fA"F.7$F]v$!3E;B\dMH_wF27$$"3fmm;9exz9F\s$!36dC*H91XE$F27$Fbv$"3*R;lWZx#pzF[p7$Fgv$"3C>*[X0RDx(F27$F\w$"3NmfCq?Oa7F.7$$"3nm;/OAo4;F\s$"3)G!4Qy$HtS"F.7$Faw$"3#3/Fu>6q]"F.7$$"3"o;a8;5Bk"F\s$"3G+nXN')fQ:F.7$$"3_L$ek8'=`;F\s$"3nmE^oz>f:F.7$$"3C+Dc6@1k;F\s$"3fZVJCS$)p:F.7$Ffw$"3`@P8hbir:F.7$F[x$"3It`ZS?bD:F.7$F`x$"3b&Q^*)em;W"F.7$$"3zLLeMPtq<F\s$"3je3Vwi^.9F.7$Fex$"3f))3+Cnuw8F.7$$"3_++v$*>04=F\s$"3$))H'yRw4n8F.7$Fjx$"3"p'fG[\Lz8F.7$Fdjo$"3#)eJ%penP]"F.7$F_y$"3e=e1W`vs<F.7$F[\p$"3VV-U'3p*p@F.7$Fdy$"3]L$H5W(R\EF.7$Fiy$"3]-KskpaYJF.7$F^z$"3E&H)p)Q/Ve$F.7$Fhz$"3[iMNb;1$*QF.7$Fb[l$"3wR&e@G#*4-%F.7$F_]p$"3!f`id!*)H@SF.7$Fd]p$"3Y8Z\-IQ4SF.7$Fi]p$"3!f"y=UbC&)RF.7$F^^p$"3R))[CW#*)*[RF.7$Fc^p$"3Cz9qGg-TQF.7$Fg[l$"3hya^vWC)o$F.7$F[_p$"38T@Qrc)fE$F.7$F\\l$"3U(\ut0`"GFF.7$$"3^LLe%)*)))yBF\s$"3mEJZcF.(=#F.7$Fa\l$"30L,"e)=rS;F.7$$"3/++vt%*paCF\s$"32XNcoZ(y7"F.7$Ff\l$"3J*4&=_:GxnF27$F`]l$!3jLK;!4p!4rF[p7$Fj]l$!3!z!RHg9x)o%F27$Fd^l$!33_`[G1&Ru(F27$F^_l$!3tI!z(ogk^7F.7$Fc_l$!3='QPv6KUj"F.7$Fh_l$!3LTeIbZ*e2#F.7$F]`l$!3!ew>JX\7a#F.7$Fb`l$!3yUWUu0c%)HF.7$F\al$!3ho_mD"4@d$F.7$F`bl$!3$4$)fe_l1v$F.7$Fdcl$!3#H%GF?"eaV$F.7$F^dl$!3ix\X'*H9`FF.7$Fcdl$!3AM$o9R:(3?F.7$Fhdl$!3U&>/bX3$\8F.7$Fbel$!3e'fOpe5R$yF27$F\fl$!3x7t7*H"3OBF27$Fffl$"3-[BwVM;xPF27$F`gl$"3)*o#yG7*QK6F.7$Fegl$"3cNBK$o%yy:F.7$Fjgl$"3/zVm'3Sj-#F.7$F_hl$"344%)\p44\CF.7$Fdhl$"3mZL(>,c5#GF.7$F^hp$"3R7r^=Z&)HJF.7$Fihl$"3%3zc4jCFM$F.7$F[ip$"3Xen?\h_5MF.7$F`ip$"3k?'yDhqCX$F.7$Feip$"3-lV=O<FpMF.7$F^il$"3XC#>bY3AY$F.7$Fbjl$"3Qcd5_dfzKF.7$F\[m$"3Q#Q#4%RL,$HF.7$Fa[m$"3.*\-4#G2tCF.7$Ff[m$"3iLf%RRXo)>F.7$F`\m$"3[K')*='QhJ9F.7$Fj\m$"3.n2C:W]/wF27$F_]m$"3WNfFCf#yx$F27$Fd]m$!3[<]Yq+g8QF[p7$Fi]m$!3"zKz2:[))y%F27$F^^m$!3))H8^bsf9$*F27$Fh^m$!3]QiVZ0ez<F.7$Fb_m$!3/]qP6;l!\#F.7$Fg_m$!3cWq]\KK<IF.7$F\`m$!35EVC%=FKC$F.7$Fa`m$!3,?G,R2,dKF.7$Ff`m$!3YT9$=]v!fKF.7$F[am$!3^z)o3%yM]KF.7$F`am$!3(HScGCA<B$F.7$Fjam$!3n@+x#y$)z;$F.7$Fdbm$!3BtxI(*G9tIF.7$Fibm$!3#>I`[L]wx#F.7$F^cm$!3K+Z'Qeu$[BF.7$Fhcm$!3NV,eZJ)>#=F.7$Fbdm$!3>\k8X%*fc6F.7$Fgdm$!3_sT'[;*e:uF27$F\em$!3\h:D,3x1IF27$Faem$"3t*4^pg<Ya"F27$Ffem$"3KNeP89I1hF27$F`fm$"3_uxn(fh\U"F.7$Fjfm$"3N-(3E^T)=@F.7$F_gm$"3F*4(=i=6nEF.7$Fdgm$"37:*yj"4-CIF.7$Figm$"3#**H.@P7D8$F.7$F^hm$"30*f/[K$H+KF.7$Fchm$"3uV5]>tL>KF.7$Fhhm$"36Y/;steGKF.7$F]im$"3#)y*z`SO!GKF.7$Fbim$"3;+;ba.h<KF.7$Fgim$"33nj10Q(G1$F.7$F\jm$"3dz'y!=tt;FF.7$Ffjm$"3AyV^\.N&>#F.7$F`[n$"3Dh(>/JIw]"F.7$Fe[n$"3#Rq&H<1]16F.7$Fj[n$"3]wP3,Q#\$oF27$F_\n$"3Xk,M&RLo[#F27$Fd\n$!3)>r[<fgp(=F27$F^]n$!3gP.R!e$*4-"F.7$Fh]n$!35>?p!p2Kw"F.7$F]^n$!3"))4dbC_FL#F.7$Fb^n$!3/?jRHE^yFF.7$Fg^n$!3Bm0_FJEBJF.7$F\_n$!3yvq1xH2sKF.7$Fa_n$!31WdQu4BwKF.7$Ff_n$!3I17$yk9#pKF.7$F[`n$!317Zf](>3D$F.7$F``n$!3*Gc2-M")3A$F.7$Fe`n$!3b>tomp'f7$F.7$Fj`n$!3fqt+l%yU)HF.7$F_an$!3w,6b[s*=`#F.7$Fdan$!3:&eH60'*3!>F.7$F^bn$!3'euBqtef<"F.7$Fhbn$!3V-o5DOvwQF27$F]cn$"3Yzi)oH()fI%F[p7$Fbcn$"3"[1=iIL**o%F27$Fgcn$"3SS`.PaxM))F27$F\dn$"3/fC)z+g1G"F.-Fadn6&FcdnFgdnFgdnFddn-%+AXESLABELSG6$Q"t6"Q!Fabt-%%VIEWG6$;FgdnF\dn;$!1#)e*yRs0\%!#:$"1utO36I?]Fjbt

Note that the solution "converges" to the staedy state as NiNmKjYjJSJ0RzciNiQlKW9wZXJhdG9yRyUmYXJyb3dHNiIlKWluZmluaXR5R0YqRipGKg==. Also note that the steady state is periodic with period NiMqKCIjPyIiIiUjUGlHRiUiIiQhIiI= which is the same as the period of the forcing function. Moreover, we observe that the steady state is not in phase with the forcing. To determine the phase difference between the steady state and forcing, we write the steady state in phase-amplitude form.s0:=simplify(subs(t=0,steady));

NiM+SSNzMEc2IiMiJytvViInQEY4

phase_amp:=A*cos(3/10*t-delta);

NiM+SSpwaGFzZV9hbXBHNiIqJkkiQUdGJSIiIi1JJGNvc0c2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliR0YlNiMsJkkidEdGJSMhIiQiIzVJJmRlbHRhR0YlRihGKA==

p0:=simplify(subs(t=0, phase_amp));

NiM+SSNwMEc2IiomSSJBR0YlIiIiLUkkY29zRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiU2I0kmZGVsdGFHRiVGKA==

s1:=simplify(subs(t=0,diff(steady,t)));

NiM+SSNzMUc2IiMiJSthIidARjg=

p1:=simplify(subs(t=0, diff(phase_amp,t)));

NiM+SSNwMUc2IiwkKiZJIkFHRiUiIiItSSRzaW5HNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJTYjSSZkZWx0YUdGJUYpIyIiJCIjNQ==

solve({p0=s0,p1=s1},{A,delta});

NiM8JC9JJmRlbHRhRzYiLUknYXJjdGFuRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHRiY2JCwkLUknUm9vdE9mR0YpNiQsJiEiIiIiIiokSSNfWkdGKSIiIyInQEY4L0kmbGFiZWxHRilJJF9MN0dGJiIjOiwkRi4iJGskL0kiQUdGJiwkRi4iJSs3

allvalues(%);

NiQ8JC9JJmRlbHRhRzYiLUknYXJjdGFuRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHRiY2IyMiIzoiJGskL0kiQUdGJiwkKiQiJ0BGOCMiIiIiIiMjIiUrN0Y0PCQvRjEsJEYzIyElKzdGNC9GJSwmRidGNkkjUGlHRiohIiI=

evalf(%);

NiQ8JC9JIkFHNiIkIislbzJSSCQhIiovSSZkZWx0YUdGJiQiK14pWyY9VCEjNjwkL0YrJCErbHJTK0pGKS9GJSQhKyVvMlJIJEYp

The steady state is displaced in phase by about 0.0412 from the forcing.Next we show that the amplitude and phase of the steady state are independent of the initial conditions (hence the initial conditions affect only the transient). To do this we find a general solution of the equation and show that NiMlInhH(0)=NiMmJSJ4RzYjIiIh and NiMqJiUjZHhHIiIiJSNkdEchIiI=(0)=NiMmJSJ4RzYjIiIi appear only in the transient terms.dsolve({subs(omega=3/10,de),x(0)=x_0,D(x)(0)=x_1},x(t));

NiMvLUkieEc2IjYjSSJ0R0YmLCoqKi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YuSShfc3lzbGliR0YmNiMsJEYoIyEiIiIjOyIiIi1JJHNpbkdGLTYjLCQqJiIkYiMjRjUiIiNGKEY1I0Y1RjRGNUY7RjwsKEkkeF8xR0YmIihPTjcjIScrS19GNUkkeF8wR0YmIidARjhGNSNGNSIpYlElUSQqKEYrRjUtSSRjb3NHRi1GOEY1LCZGQ0Y1IyEnK29WRkRGNUY1RjUtRjc2IywkRigjIiIkIiM1IyImKyE9RkQtRklGTiMiJytvVkZE

combine(%,trig):collect(%,exp);

NiMvLUkieEc2IjYjSSJ0R0YmLCgqJiwsKigtSSRzaW5HNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJjYjLCQqJiIkYiMjIiIiIiIjRihGNyNGNyIjO0Y3RjVGNkkkeF8xR0YmRjcjRjpGNSomRi1GN0Y1RjYjISYhKVskIihkaUQjKihGLUY3RjVGNkkkeF8wR0YmRjcjRjdGNSomLUkkY29zR0YvRjJGN0ZCRjdGN0ZFIyEnK29WIidARjhGNy1JJGV4cEdGLzYjLCRGKCMhIiJGOkY3RjctRi42IywkRigjIiIkIiM1IyImKyE9RkktRkZGUSMiJytvVkZJ

We now consider the general equation where NiMlJm9tZWdhRw== is a fixed positive constant. We solve the equation with that initial conditions NiMlInhH(0)=NiMmJSJ4RzYjIiIh and NiMqJiUjZHhHIiIiJSNkdEchIiI=(0)=NiMmJSJ4RzYjIiIi. We again identify the steady state and show it is independent of NiMmJSJ4RzYjIiIh and NiMmJSJ4RzYjIiIi.de;

NiMvLCgtSSVkaWZmR0kqcHJvdGVjdGVkR0YnNiQtSSJ4RzYiNiNJInRHRistSSIkR0YnNiRGLSIiIyIiIi1GJjYkRilGLSNGMiIiKUYpRjIsJC1JJGNvc0c2JEYnSShfc3lzbGliR0YrNiMqJkkmb21lZ2FHRitGMkYtRjIiIiQ=

dsolve({de,x(0)=x_0,D(x)(0)=x_1},x(t));

NiMvLUkieEc2IjYjSSJ0R0YmLCgqLC1JJGV4cEc2JEkqcHJvdGVjdGVkR0YuSShfc3lzbGliR0YmNiMsJEYoIyEiIiIjOyIiIi1JJHNpbkdGLTYjLCQqJiIkYiMjRjUiIiNGKEY1I0Y1RjRGNUY7RjwsKCIja0Y1KiRJJm9tZWdhR0YmRj0hJEYiKiRGQiIiJUZARjMsMiomSSR4XzFHRiZGNUZCRkUiJUM1ISQjPkY1KiZJJHhfMEdGJkY1RkJGRUZAKiZGSEY1RkJGPSElSz9GQUZKKiZGTEY1RkJGPUZDRkhGSUZMRkBGNSNGNUY7KipGK0Y1LUkkY29zR0YtRjhGNUY/RjMsLEZMRkBGT0ZDRktGQEZKRjVGQSIkIz5GNUY1KiYsKComLUY3NiMqJkZCRjVGKEY1RjVGQkY1IiNDLUZTRlpGVSomRkJGPUZnbkY1RkpGNUY/RjNGNQ==

combine(%,trig):gen_sol:=collect(rhs(%),exp);

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

steady:=simplify(remove(has,%,exp));

NiM+SSdzdGVhZHlHNiIsJComLCgqJi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YtSShfc3lzbGliR0YlNiMqJkkmb21lZ2FHRiUiIiJJInRHRiVGMkYyRjFGMiEiIi1JJGNvc0dGLEYvISIpKiZGMSIiI0Y1RjIiIilGMiwoIiNrRjIqJEYxRjkhJEYiKiRGMSIiJUY8RjQhI0M=

We now write the steady state in the phase amplitude form.s0:=simplify(subs(t=0,steady));

NiM+SSNzMEc2IiwkKiYsJiEiIiIiIiokSSZvbWVnYUdGJSIiI0YqRiosKCIja0YqRishJEYiKiRGLCIiJUYvRikhJCM+

phase_amp:=A*cos(omega*t-delta);

NiM+SSpwaGFzZV9hbXBHNiIqJkkiQUdGJSIiIi1JJGNvc0c2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliR0YlNiMsJiomSSZvbWVnYUdGJUYoSSJ0R0YlRihGKEkmZGVsdGFHRiUhIiJGKA==

p0:=simplify(subs(t=0,phase_amp));

NiM+SSNwMEc2IiomSSJBR0YlIiIiLUkkY29zRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiU2I0kmZGVsdGFHRiVGKA==

s1:=simplify(subs(t=0,diff(steady,t)));

NiM+SSNzMUc2IiwkKiZJJm9tZWdhR0YlIiIjLCgiI2siIiIqJEYoRikhJEYiKiRGKCIiJUYrISIiIiND

p1:=simplify(subs(t=0,diff(phase_amp,t)));

NiM+SSNwMUc2IiooSSJBR0YlIiIiLUkkc2luRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiU2I0kmZGVsdGFHRiVGKEkmb21lZ2FHRiVGKA==

solve({p0=s0,p1=s1},{A,delta});

NiM8JC9JJmRlbHRhRzYiLUknYXJjdGFuRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHRiY2JComSSZvbWVnYUdGJiIiIi1JJ1Jvb3RPZkdGKTYkLCYhIiJGLyomLCgiI2tGLyokRi4iIiMhJEYiKiRGLiIiJUY3Ri9JI19aR0YpRjlGLy9JJmxhYmVsR0YpSSVfTDEyR0YmRi8qJiwmIiIpRi9GOCEiKUYvRjBGLy9JIkFHRiYsJEYwIiND

allvalues(%);

NiQ8JC9JIkFHNiIsJCokKiQsKCIjayIiIiokSSZvbWVnYUdGJiIiIyEkRiIqJEYuIiIlRishIiIjRixGLyIjQy9JJmRlbHRhR0YmLUknYXJjdGFuRzYkSSpwcm90ZWN0ZWRHRjtJKF9zeXNsaWJHRiY2JComRi5GLEYpRjQqJiwmIiIpRixGLSEiKUYsRilGNDwkL0Y3LUY5NiQsJEY+RjMsJEY/RjMvRiUsJEYoISND

assign(%[1]);We take a value of NiMlJmRlbHRhRw== corresponding to a positive value for NiMlIkFH. We now compute the phase lag for different NiMlJm9tZWdhRw==.evalf(subs(omega=1,delta));

NiMkIitGanpxOiEiKg==

evalf(subs(omega=2,delta));

NiMkIitBOVhlSSEiKg==

evalf(subs(omega=3,delta));

NiMkIitUPnYlNCQhIio=

So the phase lag is approximately 1.571 when NiMvJSZvbWVnYUciIiI=, 3.058 when NiMvJSZvbWVnYUciIiM=, and 3.095 when NiMvJSZvbWVnYUciIiQ=.plot(delta,omega=0.1..10);

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

We also determine the amplitude of the steady state as a function of the forcing frequency NiMlJm9tZWdhRw==.A;

NiMsJCokKiQsKCIjayIiIiokSSZvbWVnYUc2IiIiIyEkRiIqJEYqIiIlRichIiIjRihGLCIjQw==

plot(A,omega=0.1..10);

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

We finally find the value of NiMlJm9tZWdhRw== for which the amplitude is maximal.Aprime:=diff(A,omega);

NiM+SSdBcHJpbWVHNiIsJCooKiQsKCIjayIiIiokSSZvbWVnYUdGJSIiIyEkRiIqJEYtIiIlRiohIiIjRjJGLkYpISIjLCZGLSEkYSMqJEYtIiIkIiRjI0YrISM3

solve(Aprime=0,omega);

NiUiIiEsJCokIiRhIyMiIiIiIiMjRigiIzssJEYlIyEiIkYr

evalf(%);

NiUkIiIhRiQkIisxNCczJyoqISM1JCErMTQnMycqKkYn

The maximum amplitude occurs when NiMlJm9tZWdhRw== is approximately 0.9961. This value is closeto 1 at which the corresponding undamped oscillator exhibits resonance. Also not that we exclude NiMvJSZvbWVnYUciIiE= since then the forcing is constant.subs(omega=.9960860906,A);

NiMkIistOHEvQyEiKQ==

Also note that unlike resonance the amplitude for this NiMlJm9tZWdhRw== is still finite (approximately 24.05).