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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "(ARKISTO,OPISKELU,MATEMATI
IKKA,DIFFERENTIAALIYHT\304L\326T)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 18 "" 0 "" {TEXT -1 35 "DIFFERENTIAALIYHT\304L\326IDEN K\304SITT
EIT\304" }}{PARA 19 "" 0 "" {TEXT -1 19 "by Juha L\344hteenm\344ki" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "
Differentiaaliyht\344l\366(=DY) " }{TEXT 260 10 "M\304\304RITELM\304" 
}}{PARA 0 "" 0 "" {TEXT 257 20 "DIFFERENTIAALIYHT\304L\326" }}{PARA 0 
"" 0 "" {TEXT -1 208 "Differentiaaliyht\344l\366 (DY) on yht\344l\366,
 joka sis\344lt\344\344 riippumattoman muuttujan esim: x tai t, tuntem
attoman funktion esim: y=y(x) tai y=y(t) ja sen derivaattoja. DY kuvaa
 usein jonkin j\344rjestelm\344n k\344ytt\344ytymist\344" }}{PARA 0 "
" 0 "" {TEXT -1 22 "esim. ajan (t) suhteen" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Esim1: " }{XPPEDIT 18 0 "a3(x)*d
iff(y(x),x,x,x)+a2(x)*diff(y(x),x)+a1(x)*y(x)=3*x^2+2*x+7" "/,(*&-%#a3
G6#%\"xG\"\"\"-%%diffG6&-%\"yG6#F(F(F(F(F)F)*&-%#a2G6#F(F)-F+6$-F.6#F(
F(F)F)*&-%#a1G6#F(F)-F.6#F(F)F),(*&\"\"$F)*$F(\"\"#F)F)*&FBF)F(F)F)\"
\"(F)" }{TEXT -1 2 "  " }{TEXT 258 88 "on differentiaali yht\344l\366 \+
sama voitaisiin my\366s merkit\344 a3(x)y'''(x)+a2(x)y'(x)+a1(x)y(x)=
" }{XPPEDIT 18 0 "3*x^2+2*x+7" ",(*&\"\"$\"\"\"*$%\"xG\"\"#F%F%*&F(F%F
'F%F%\"\"(F%" }{TEXT 259 34 " (vrt.derivaattojen merkitseminen)" }}
{PARA 0 "" 0 "" {TEXT -1 9 "Ks. my\366s " }{HYPERLNK 17 "differentiaal
iyht\344l\366t " 1 "differentiaaliyht\344l\366.mws" "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 136 "J\344rjestel
m\344 analysointiin liittyv\344t DY:n k\344sitteet : sis\344\344nmeno=
 her\344te = input ja ulostulo = vaste= output ja siirtofunktio sek
\344 karpo." }}{PARA 3 "" 0 "" {TEXT 342 216 " Sis\344\344nmenolla tar
koitetaan h\344iri\366funktiota siis funktiota joka DY:n normaalimuodo
ssa on yksin\344\344n = merkin oikealla puolella. Ulostulo taas on se \+
funktio joka derivaattoineen on samaisen merkin vasemmalla puolella.  \+
" }}{PARA 3 "" 0 "" {TEXT 343 50 "Esim. Lineaarinen DY voidaan aina an
taa muodossa: " }}{PARA 0 "" 0 "" {XPPEDIT 332 0 "y^n" ")%\"yG%\"nG" }
{TEXT 333 4 " + a" }{TEXT 335 3 "n-1" }{TEXT 336 3 "(x)" }{XPPEDIT 
338 0 "y^(n-1)" ")%\"yG,&%\"nG\"\"\"F&!\"\"" }{TEXT 337 1 "+" }{TEXT 
-1 1 " " }{TEXT 339 33 " ... + a1(x) y' + a0(x) y = f(x) " }{TEXT -1 
8 ", miss\344 " }{XPPEDIT 18 0 "y^n" ")%\"yG%\"nG" }{TEXT -1 49 " on y
:n n:nen kertaluvun derivaatta x:n suhteen, " }{TEXT 334 1 "a" }{TEXT 
340 3 "n-1" }{TEXT 341 11 "(x)...a0(x)" }{TEXT -1 99 " ovat DY:n kerro
in funktiot ja y(x) siis ulostulo eli vaste ja f(x) h\344iri\366funkti
o eli sis\344\344nmeno.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 422 "M\344\344ritell\344\344n Siirtofunktio H(s) on si
s\344\344nmenon Laplace-muunnoksen U(s) kerroin  , kun ulostulon Lapla
ce-muunnos Y(s) on ratkaistuna. Karakteristinen polynomi eli karpo taa
s vastaa karakteristisen yht\344l\366n vasenta puolta jos karakteristi
sen muuttujan (yleens\344 r) paikalle on sijoitettu Laplace-muuttuja s
. Eli karpo on Laplace-muunnetun DY:n Y(s):n kerroin. Monesti siis H(s
) = 1 / karpo. Seuraava esimerkki selvent\344\344 asiaa." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Esim. Olkoon j\344
rjestelm\344\344 kuvaava DY muotoa:  y''(t) + 3y'(t) + 2y(t) = u(t) .
" }}{PARA 0 "" 0 "" {TEXT -1 43 "a) Mik\344 on ko. DY:n sis\344\344nme
no ja ulostulo?" }}{PARA 0 "" 0 "" {TEXT -1 44 "Vastaus: sis\344\344nm
eno on u(t) ja ulostulo y(t)" }}{PARA 0 "" 0 "" {TEXT -1 39 "b) Mik
\344 on DY:n siirtofunktio ja karpo?" }}{PARA 0 "" 0 "" {TEXT -1 34 "K
arakteristinen yht\344l\366 on muotoa: " }{XPPEDIT 18 0 "r^2" "*$%\"rG
\"\"#" }{TEXT -1 31 " + 3r + 2 = 0  joten karpo on  " }{XPPEDIT 18 0 "
s^2" "*$%\"sG\"\"#" }{TEXT -1 10 " + 3s + 2 " }}{PARA 0 "" 0 "" {TEXT 
-1 24 "Laplace-muunnettu DY: ( " }{XPPEDIT 18 0 "s^2" "*$%\"sG\"\"#" }
{TEXT -1 43 " + 3s + 2) Y(s) = U(s) + alkuarvotermi --> " }}{PARA 0 "
" 0 "" {TEXT -1 7 "Y(s) = " }{XPPEDIT 18 0 "U(s)/(s^2+3*s+2)" "*&-%\"U
G6#%\"sG\"\"\",(*$F&\"\"#F'*&\"\"$F'F&F'F'F*F'!\"\"" }{TEXT -1 3 " + \+
" }{XPPEDIT 18 0 "alkuarvotermi/(s^2+3*s+2)" "*&%.alkuarvotermiG\"\"\"
,(*$%\"sG\"\"#F$*&\"\"$F$F'F$F$F(F$!\"\"" }{TEXT -1 42 "   t\344st\344
 seuraa ett\344 siirtofunktio H(s) = " }{XPPEDIT 18 0 "1/(s^2+3*s+2)" 
"*&\"\"\"F#,(*$%\"sG\"\"#F#*&\"\"$F#F&F#F#F'F#!\"\"" }{TEXT -1 3 " = \+
" }{XPPEDIT 18 0 "1/karpo" "*&\"\"\"F#%&karpoG!\"\"" }{TEXT -1 2 "  " 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 
19 "Vakiokertoiminen DY" }}{PARA 0 "" 0 "vakiokertoiminen DY" {TEXT 
261 19 "VAKIOKERTOIMINEN DY" }}{PARA 0 "" 0 "" {TEXT -1 24 "Lineaarine
n DY on muotoa" }}{PARA 0 "" 0 "" {XPPEDIT 346 0 "y^n" ")%\"yG%\"nG" }
{TEXT 347 4 " + a" }{TEXT 349 3 "n-1" }{TEXT 350 3 "(x)" }{XPPEDIT 
352 0 "y^(n-1)" ")%\"yG,&%\"nG\"\"\"F&!\"\"" }{TEXT 351 1 "+" }{TEXT 
-1 1 " " }{TEXT 353 33 " ... + a1(x) y' + a0(x) y = f(x) " }{TEXT -1 
8 ", miss\344 " }{XPPEDIT 18 0 "y^n" ")%\"yG%\"nG" }{TEXT -1 49 " on y
:n n:nen kertaluvun derivaatta x:n suhteen, " }{TEXT 348 1 "a" }{TEXT 
354 3 "n-1" }{TEXT 355 11 "(x)...a0(x)" }{TEXT -1 28 " ovat DY:n kerro
in funktiot " }}{PARA 0 "" 0 "" {TEXT -1 37 "DY on vakiokertoiminen jo
s funktiot a" }{TEXT 356 3 "n-1" }{TEXT -1 55 "(x),...,a1(x), miss\344
 n on DY:n asteluku, ovat vakioita." }}{PARA 0 "" 0 "" {TEXT -1 9 "ks.
 my\366s " }{HYPERLNK 17 "vakiokertoimisen DY:n ratkaisu" 1 "different
iaaliyht\344l\366.mws" "vakiokertoimisen DY:n ratkaisu" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Differentiaa
liyht\344l\366ryhm\344" }}{PARA 0 "" 0 "" {TEXT 262 25 "DIFFERENTIAALI
YHT\304L\326RYHM\304" }}{PARA 0 "" 0 "" {TEXT -1 126 "Sis\344lt\344
\344 riippumattoman muuttujan t ja tuntemattomia funktiota x1=x1(t), x
2=x2(t), x3=x3(t)...xn=xn(t) ja n\344iden derivaattoja." }}{PARA 256 "
" 0 "" {TEXT -1 19 "Esim2: x1'(t)=x2(t)" }}{PARA 257 "" 0 "" {TEXT -1 
23 "           x2'(t)=x3(t)" }}{PARA 258 "" 0 "" {TEXT -1 28 "        \+
   x3'(t)=-3x2(t)+2t" }}{PARA 0 "" 0 "" {TEXT -1 12 "on DY ryhm\344." 
}}{PARA 0 "" 0 "" {TEXT -1 8 "Ks my\366s " }{HYPERLNK 17 "DY-ryhm\344n
 ratkaiseminen" 1 "differentiaaliyht\344l\366.mws" "useita differentia
aliyht\344l\366it\344" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "Osittaisdiff
erentiaaliyht\344l\366" }}{PARA 0 "" 0 "" {TEXT 263 28 "OSITTAISDIFFER
ENTIAALIYHT\304L\326" }}{PARA 0 "" 0 "" {TEXT -1 183 "Osittaisdifferen
tiaaliyht\344l\366 sis\344lt\344\344 useita riippumattomattomia muuttu
jia x1...xn tai t1...tn ja tuntemattoman funktion y=y(x1,...,xn) tai y
=y(t1...tn) ja t\344m\344n osittaisderivaattoja." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 
"" {TEXT -1 18 "DY:n merkitseminen" }}{PARA 0 "" 0 "" {TEXT 264 13 "ME
RKITSEMINEN" }}{PARA 0 "" 0 "" {TEXT -1 30 "ks derivaattojen merkitsem
inen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "DY:n kertaluku" }}{PARA 0 "" 
0 "" {TEXT 265 9 "KERTALUKU" }}{PARA 0 "" 0 "" {TEXT -1 60 "DY:ss\344 \+
esiintyv\344n tuntemattoman funktion korkein derivaatta." }}{PARA 0 "
" 0 "" {TEXT 266 31 "Esim3: esimerkiksi esimerkiss\344 " }{XPPEDIT 18 
0 "a3(x)*diff(y(x),x,x,x)+a2(x)*diff(y(x),x)+a1(x)*y(x)=3*x^2+2*x+7" "
/,(*&-%#a3G6#%\"xG\"\"\"-%%diffG6&-%\"yG6#F(F(F(F(F)F)*&-%#a2G6#F(F)-F
+6$-F.6#F(F(F)F)*&-%#a1G6#F(F)-F.6#F(F)F),(*&\"\"$F)*$F(\"\"#F)F)*&FBF
)F(F)F)\"\"(F)" }{TEXT -1 1 " " }{TEXT 331 22 "  DY:n kertaluku on 3.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "DY:n ratkaisu" }}{PARA 0 "" 0 "" 
{TEXT 267 8 "RATKAISU" }}{PARA 0 "" 0 "" {TEXT -1 45 "Funktio joka tot
euttaa DY:n jollakin v\344lill\344." }}{PARA 0 "" 0 "" {TEXT -1 3 "ks.
" }{HYPERLNK 17 "DY:n ratkaiseminen" 1 "differentiaaliyht\344l\366.mws
" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "DY:n lineaarisuus ja ep\344line
aarisuus" }}{PARA 0 "" 0 "lineaarinen DY" {TEXT 268 14 "LINEAARINEN DY
" }}{PARA 0 "" 0 "" {TEXT -1 89 "DY:t jaetaan lineaarisiin ja ep\344li
neaarisiin.Lineaarinen DY voidaan aina antaa muodossa: " }}{PARA 0 "" 
0 "" {XPPEDIT 293 0 "y^n" ")%\"yG%\"nG" }{TEXT 294 4 " + a" }{TEXT 
296 3 "n-1" }{TEXT 297 3 "(x)" }{XPPEDIT 299 0 "y^(n-1)" ")%\"yG,&%\"n
G\"\"\"F&!\"\"" }{TEXT 298 1 "+" }{TEXT -1 1 " " }{TEXT 300 33 " ... +
 a1(x) y' + a0(x) y = f(x) " }{TEXT -1 8 ", miss\344 " }{XPPEDIT 18 0 
"y^n" ")%\"yG%\"nG" }{TEXT -1 49 " on y:n n:nen kertaluvun derivaatta \+
x:n suhteen, " }{TEXT 295 1 "a" }{TEXT 301 3 "n-1" }{TEXT 302 11 "(x).
..a0(x)" }{TEXT -1 118 " ovat DY:n kerroin funktiot ja f(x) h\344iri
\366funktio eli sis\344\344nmeno. Huom. kerroin funktio voi tietenkin \+
olla nollakin.) " }}{PARA 0 "" 0 "" {TEXT -1 552 "J\344rjestelm\344 an
alyysiss\344 sanotaan j\344rjestelm\344n olevan lineaarinen kun sit
\344 kuvaava DY tai Differenssiyht\344l\366 on lineaarinen eli kun vas
te=ulostulo on lineaarinen. Eli ulostulo y(t) piirrettyn\344 sis\344
\344nmenon (yleens\344  j\344rjestelm\344 analyysiss\344 u(t) ) funkti
ona antaa siis t\344ss\344tapauksessa k\344yr\344ksi suoran. Yleens
\344 t\344m\344 kannattaa voidaan todentaa kuvallisesti Laplace-muunta
malla DY ja piirt\344m\344ll\344 ulostulon Laplace-muunnos sis\344\344
nmenon Laplace-muunnoksen funktiona. Luonnollisesti lineaarisuuden m
\344\344ritelm\344\344 voidaan soveltaa DY:t\344 lineaariseksi osoitet
taessa. " }}{PARA 0 "" 0 "" {TEXT -1 72 "DY ryhm\344 on ep\344lineaari
nen jos yksikin sen yht\344l\366ist\344 on ep\344lineaarinen." }}
{PARA 259 "" 0 "" {TEXT -1 7 "Esim4: " }{TEXT 269 17 "y'''(x)+3y'(x) -
 " }{XPPEDIT 18 0 "5*x^2" "*&\"\"&\"\"\"*$%\"xG\"\"#F$" }{TEXT 312 6 "
y(x)=0" }{TEXT -1 4 " ,  " }{TEXT 270 9 "y''(x)+8 " }{XPPEDIT 18 0 "e^
(2x^2" ")%\"eG*&\"\"#\"\"\"*$%\"xGF%F&" }{TEXT 314 5 "y(x)=" }
{XPPEDIT 18 0 "x^5" "*$%\"xG\"\"&" }{TEXT -1 9 "+sin(x), " }{XPPEDIT 
18 0 "a3(x)*diff(y(x),x,x,x)+a2(x)*diff(y(x),x)+a1(x)*y(x)=3*x^2+2*x+7
" "/,(*&-%#a3G6#%\"xG\"\"\"-%%diffG6&-%\"yG6#F(F(F(F(F)F)*&-%#a2G6#F(F
)-F+6$-F.6#F(F(F)F)*&-%#a1G6#F(F)-F.6#F(F)F),(*&\"\"$F)*$F(\"\"#F)F)*&
FBF)F(F)F)\"\"(F)" }{TEXT -1 1 " " }{TEXT 315 23 " ja y''(x)+y'(x)+y(x
)=0" }{TEXT -1 1 " " }{TEXT 313 30 "sek\344  sin(x )y'(x) + y(x) = 0 \+
" }{TEXT -1 16 "ovat lineaarisia" }{TEXT 311 11 " sensijaan " }{TEXT 
-1 12 " y(x)y'''(x)" }{TEXT 316 10 "+3y'(x) - " }{XPPEDIT 18 0 "5*x^2
" "*&\"\"&\"\"\"*$%\"xG\"\"#F$" }{TEXT 318 6 "y(x)=0" }{TEXT -1 4 " , \+
 " }{TEXT 317 9 "y''(x)+8 " }{XPPEDIT 18 0 "e^(2x^2" ")%\"eG*&\"\"#\"
\"\"*$%\"xGF%F&" }{XPPEDIT 18 0 "y(x)^3" "*$-%\"yG6#%\"xG\"\"$" }
{TEXT 319 1 "=" }{XPPEDIT 18 0 "x^5" "*$%\"xG\"\"&" }{TEXT -1 9 "+sin(
x), " }{XPPEDIT 18 0 "a3(x)*diff(y(x),x,x,x)+a2(x)*diff(y(x),x)^2+a1(x
)*y(x)=3*x^2+2*x+7" "/,(*&-%#a3G6#%\"xG\"\"\"-%%diffG6&-%\"yG6#F(F(F(F
(F)F)*&-%#a2G6#F(F)*$-F+6$-F.6#F(F(\"\"#F)F)*&-%#a1G6#F(F)-F.6#F(F)F),
(*&\"\"$F)*$F(F9F)F)*&F9F)F(F)F)\"\"(F)" }{TEXT -1 1 " " }{TEXT 320 
17 " ja y''(x)+y'(x)+" }{XPPEDIT 18 0 "sqrt(y(x))" "-%%sqrtG6#-%\"yG6#
%\"xG" }{TEXT 322 2 "=0" }{TEXT -1 1 " " }{TEXT 321 27 "sek\344 sin(y'
(x)) + y(x) = 0 " }{TEXT -1 22 "eiv\344t ole lineaarisia." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "ep\344lineaarinen DY" {TEXT 271 
17 "EP\304LINEAARINEN DY" }}{PARA 0 "" 0 "" {TEXT -1 43 "DY joka ei ol
e lineaarinen. Ks. edel. esim." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "D
Y:n normaalimuoto" }}{PARA 0 "" 0 "" {TEXT 272 18 "DY:n NORMAALIMUOTO
" }}{PARA 0 "" 0 "" {TEXT -1 198 "DY:n normaalimuoto on DY kirjoitettu
na muodossa, jossa vasemmalla puolella ovat DY:n tuntematon funktio es
im y(x) tai y(t) ja kaikki t\344m\344n funktion derivaatat ja oikealla
 puolella kaikki muu tavara." }}{PARA 0 "" 0 "" {TEXT -1 150 "Lis\344k
si korkeimman derivaatan kertoimena tulisi yleens\344 olla 1. T\344m
\344 saadaan helposti aikaan kun jaetaan puolittain korkeimman derivaa
tan kertoimella." }}{PARA 0 "" 0 "" {TEXT -1 16 "Esimerkiksi DY :" }
{TEXT 307 1 " " }{XPPEDIT 18 0 "y^n" ")%\"yG%\"nG" }{TEXT -1 4 " + a" 
}{TEXT 304 3 "n-1" }{TEXT -1 3 "(x)" }{XPPEDIT 18 0 "y^(n-1)" ")%\"yG,
&%\"nG\"\"\"F&!\"\"" }{TEXT -1 1 "+" }{TEXT 303 1 " " }{TEXT -1 55 " .
.. + a1(x) y' + a0(x) y = f(x) on normaali muodossa. " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 30 "Esim5: esimerkin 1 ja
 4 DY:t (" }{XPPEDIT 18 0 "a3(x)*diff(y(x),x,x,x)+a2(x)*diff(y(x),x)+a
1(x)*y(x)=3*x^2+2*x+7" "/,(*&-%#a3G6#%\"xG\"\"\"-%%diffG6&-%\"yG6#F(F(
F(F(F)F)*&-%#a2G6#F(F)-F+6$-F.6#F(F(F)F)*&-%#a1G6#F(F)-F.6#F(F)F),(*&
\"\"$F)*$F(\"\"#F)F)*&FBF)F(F)F)\"\"(F)" }{TEXT -1 1 " " }{TEXT 310 
47 " ja y''(x)+y'(x)+y(x)=0) ovat normaalimuodossa." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 
0 "h\344iri\366funktio" {TEXT -1 34 "DY:n h\344iri\366funktio eli sis
\344\344nmeno." }}{PARA 0 "" 0 "" {TEXT 274 13 "H\304IRI\326FUNKTIO" }
}{PARA 0 "" 0 "" {TEXT -1 103 "H\344iri\366funktio on funktio joka sij
aitsee DY:n oikealla puolella kun DY on kirjoitettu normaalimuotoonsa.
" }}{PARA 0 "" 0 "" {TEXT -1 97 "Varsinkin aikariippuvilla j\344rjeste
lmill\344 h\344iri\366funktiosta k\344ytet\344\344n usein nimityst\344
 sis\344\344nmeno.  " }}{PARA 0 "" 0 "" {TEXT -1 15 " Esim. DY:n    " 
}{XPPEDIT 18 0 "y^n" ")%\"yG%\"nG" }{TEXT -1 4 " + a" }{TEXT 305 3 "n-
1" }{TEXT -1 3 "(x)" }{XPPEDIT 18 0 "y^(n-1)" ")%\"yG,&%\"nG\"\"\"F&!
\"\"" }{TEXT 306 1 "+" }{TEXT -1 55 "  ... + a1(x) y' + a0(x) y = f(x)
 h\344iri\366funktio on f(x)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 275 54 "Esim6: y'(x)+y(x)=3x+2  funktion h\344iri\366f
unktio on 3x+2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 47 "H
omogeeninen DY(=HY) ja ep\344homogeeninen DY(=EY)" }}{PARA 0 "" 0 "hom
ogeeninen DY" {TEXT 276 18 "DY:n HOMOGEENISUUS" }}{PARA 0 "" 0 "" 
{TEXT 308 36 "DY on homogeeninen jos h\344iri\366funktio" }{TEXT -1 1 
" " }{TEXT 309 4 "on 0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "
 ( " }{TEXT 324 55 "Yleinen homogeeninen (ja lineaarinen) DY on muotoa
:    " }{XPPEDIT 325 0 "y^n" ")%\"yG%\"nG" }{TEXT 326 4 " + a" }{TEXT 
330 3 "n-1" }{TEXT 327 3 "(x)" }{XPPEDIT 328 0 "y^(n-1)" ")%\"yG,&%\"n
G\"\"\"F&!\"\"" }{TEXT 323 1 "+" }{TEXT 329 30 "  ... + a1(x) y' + a0(
x) y = 0" }}{PARA 0 "" 0 "" {TEXT 277 44 "Esim7: esimerkin 4 DY:t (y''
(x)+y'(x)+y(x)=0" }{TEXT -1 4 " ja " }{TEXT 279 13 "y''(x)+y(x)=0" }
{TEXT -1 1 " " }{TEXT 278 20 ") ovat homogeeniset." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "ep\344homogeeninen DY" {TEXT 280 21 "DY:
n EP\304HOMOGEENISUUS" }}{PARA 0 "" 0 "" {TEXT -1 57 "DY on ep\344homo
geeninen jos h\344irifunktio on erisuuri kuin 0" }}{PARA 0 "" 0 "" 
{TEXT 281 59 "Esim8: esimerkin 6 DY (y'(x)+y(x)=3x+2) on ep\344homogee
ninen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 75 "DY:n yleinen ratkaisu, yksit
yisratkaisu, AAP:n ratkaisu ja erikoisratkaisu." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 59 "DY:n YLEINEN RATKAISU , \+
YKSITYISRATKAISU JA ERIKOISRATKAISU" }}{PARA 0 "" 0 "" {TEXT 283 16 "Y
leinen ratkaisu" }{TEXT -1 160 " saadaan osaratkaisuiden (ratkaisukomp
onenttien ) lineaarikompinaationa ja sis\344lt\344\344 n kpl toisistaa
n riippumatonta parametria C1...Cn miss\344 n on DY:n kertaluku." }}
{PARA 0 "" 0 "" {TEXT 284 41 "Yksityisratkaisu=partikulaarinen ratkais
u" }{TEXT -1 65 " saadaan kun kiinnitet\344\344n parametreille C1...Cn
 m\344\344r\344tyt arvot.  " }}{PARA 0 "" 0 "AAP:n ratkaisu" {TEXT -1 
75 "Jos n\344m\344 arvot saadaan kiinnitetyksi DY:n alkuehtojen perust
eella puhutaan " }{TEXT 344 5 "AAP:n" }{TEXT -1 24 " eli alkuarvoprobl
eeman " }{TEXT 345 11 "ratkaisusta" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 38 "Erikoisratkaisu=singulaa
rinen ratkaisu" }{TEXT -1 49 " on ratkaisu jota ei saada yleisest\344 \+
ratkaisusta." }}{PARA 0 "" 0 "" {TEXT 285 12 "esim9: DY:n " }{XPPEDIT 
18 0 "(diff(y,x)^2-x*diff(y,x)+y=0" "/,(*$-%%diffG6$%\"yG%\"xG\"\"#\"
\"\"*&F)F+-F&6$F(F)F+!\"\"F(F+\"\"!" }{TEXT -1 1 " " }{TEXT 288 19 "yl
einen ratkaisu on" }{TEXT -1 1 " " }{XPPEDIT 18 0 "y(x)=C*x-C^2" "/-%
\"yG6#%\"xG,&*&%\"CG\"\"\"F&F*F**$F)\"\"#!\"\"" }{TEXT -1 1 " " }
{TEXT 287 141 "olkoon alkuehto y(0)=-1 t\344ll\366in saadaan C:lle C=1
. Eli yksityisratkaisu alkuehdolle y(0)=-1 on y=x-1. Toisaalta  DY:ll
\344 on erikoisratkaisuna " }{XPPEDIT 18 0 "y=x^2/4" "/%\"yG*&%\"xG\"
\"#\"\"%!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "DY:n verhok\344yr
\344" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 10 "
VERHOK\304YR\304" }}{PARA 0 "" 0 "" {TEXT -1 146 "Verhok\344yr\344 on \+
erikoisratkaisun kuvaaja ja se sivuaa yleisest\344 ratkaisusta eri par
ametrien arvoilla C1 ja C2 saatavan ratkaisuk\344yr\344parven kuvaajia
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 3 "" 0 "DY:n aikainkvarianttisuus" {TEXT -1 25 "DY:n aik
ainkvarianttisuus" }}{PARA 0 "" 0 "" {TEXT -1 90 "DY:n  sanotaan oleva
n aikainkvariantti jos sen mallintama j\344rjestelm\344 on aikainkvari
antti." }}{PARA 0 "" 0 "" {TEXT -1 92 "(j\344rjestelm\344 on aikainkva
riantti jos se antaa saman ulostulon aina aikahetkest\344 riippumatta)
" }}{PARA 0 "" 0 "" {TEXT -1 9 "ks.LINKKI" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 1 {PARA 3 "" 0 "DY:n karakteristinen yht\344l\366" 
{TEXT -1 27 "DY:n karakteristinen yht\344l\366" }}{PARA 0 "" 0 "" 
{TEXT -1 127 "Karakteristista yht\344l\366a tarvitaan vakiokertoimisen
 (homogeenisen ja sit\344kautta my\366s ep\344homogeenisen  DY:n ratka
isemisessa (ks. " }{HYPERLNK 17 "DY:n ratkaiseminen)" 1 "differentiaal
iyht\344l\366.mws" "miten ratkaisen differentiaaliyht\344l\366it\344" 
}{TEXT -1 183 " Karakteristisen yht\344l\366n muodostamisen periaattee
na on se ett\344 muodostetaan DY:n kertalukua vastaavan asteinen algeb
rallinen yht\344l\366 jossa jokaista n:nen kertaluvun derivaattaa vast
aa " }{XPPEDIT 18 0 "r^n" ")%\"rG%\"nG" }{TEXT -1 216 " jne. Vakiokert
oimet s\344ilyv\344t samoina.  Esim n:nen kertaluvun homogeenisen vaki
okertoimisen DY:n karakteristinen yht\344l\366 on muotoa:             \+
                                                                      \+
   " }}{PARA 0 "" 0 "" {XPPEDIT 291 0 "r^n+a(n-1)*r^(n-1)" ",&)%\"rG%
\"nG\"\"\"*&-%\"aG6#,&F%F&F&!\"\"F&)F$,&F%F&F&F,F&F&" }{TEXT 292 22 "+
.....+ a1 r + a0 = 0 " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "AAP(alk
uarvoprobleema) " }}{PARA 0 "" 0 "" {TEXT 290 23 "ALKUARVOPROBLEEMA = \+
AAP" }}{PARA 0 "" 0 "" {TEXT -1 109 "AAP:ss\344 on m\344\344r\344tt
\344v\344 DY:n se ratkaisu, joka ja jonka derivaatat ovat ennalta m
\344\344r\344ttyj\344 tietyiss\344 pisteiss\344." }}{PARA 0 "" 0 "" 
{TEXT -1 24 "vrt reuna-arvoprobleema." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 86 "Paloittain jatkuvien funktio
iden DY:t esitet\344\344n Heavisiden steppifunktion H(t) avulla." }}
{PARA 0 "" 0 "Heavisiden steppifunktio" {TEXT -1 200 "Heaviside funkti
on avulla pystyt\344\344n tarkastelemaan paloittain m\344\344riteltyj
\344 tapauksia. Heaviside funktio = H(t) on 0 kun suluissa olevan laus
ekkeen arvo on < 0. Muulloin Heavisiden funktion arvo on 1. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 199 "Ks. tarkempia \+
ohjeita: kirjasta(punakantinen) Glyn James Advanced Modern Engineerin \+
Mathematics    sivuilta 102 - 220 Ja vihre\344st\344 kansiosta joka si
s\344lt\344\344 kurssin fuorierin menetelm\344t muistiinpanoja." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 117 
"Esim. Esit\344 paloittain m\344\344ritelty funktio f(t) = t kun t < a
 ja f(t) = 2t kun t > a Heavisiden steppifunktion avulla." }}{PARA 0 "
" 0 "" {TEXT -1 29 "f(t) = H(a - t) t + H(t-a) 2t" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}}{MARK "32 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }