Boy/DiPrima/Mad h d, Ch 6.: Diniion o apla Tranorm Elmnary Dirnial Equaion and Boundary Valu Problm, h diion, by William E. Boy, Rihard C. DiPrima, and Doug Mad 7 by John Wily & Son, In. Many praial nginring problm involv mhanial or lrial ym ad upon by dioninuou or impuliv oring rm. For uh problm h mhod dribd in Chapr 3 ar diiul o apply. In hi hapr w u h apla ranorm o onvr a problm or an unknown union ino a implr problm or F, olv or F, and hn rovr rom i ranorm F. Givn a known union K,, an ingral ranorm o a union i a rlaion o h orm b F = ò K, d, - a <b a
Impropr Ingral Th apla ranorm will involv an ingral rom zro o ininiy. Suh an ingral i a yp o impropr ingral. An impropr ingral ovr an unboundd inrval i dind a h limi o an ingral ovr a ini inrval d lim a d A whr A i a poiiv ral numbr. I h ingral rom a o A xi or ah A > a and i h limi a A xi, hn h impropr ingral i aid o onvrg o ha limiing valu. Ohrwi, h ingral i aid o divrg or ail o xi. a A
Exampl Conidr h ollowing impropr ingral. d W an valua hi ingral a ollow: ò d Thror, h impropr ingral divrg. ò = lim A ò A d = lim ln A A
Exampl Conidr h ollowing impropr ingral. W an valua hi ingral a ollow: d A lim lim A d d A A No ha i =, hn =. Thu h ollowing wo a hold: d, i ; and d divrg, i.
Exampl 3 Conidr h ollowing impropr ingral. p From Exampl, hi ingral divrg a p = W an valua hi ingral or p a ollow: A p d lim p d lim A A A p Th impropr ingral divrg a p = and I p, I p, lim A lim A d p p A p p p A p
Piwi Coninuou Funion A union i piwi oninuou on an inrval [a, b] i hi inrval an b pariiond by a ini numbr o poin a = < < < n = b uh ha i oninuou on ah k, k+ lim k, k,, n 3 lim k, k,, n In ohr word, i piwi oninuou on [a, b] i i i oninuou hr xp or a ini numbr o jump dioninuii.
Thorm 6.. I i piwi oninuou or a, i g whn M or om poiiv M and i gd onvrg, hn ò M d alo onvrg. ò a On h ohr hand, i g or M, and i divrg, hn d alo divrg. ò a ò M gd
Th apla Tranorm b a union dind or >, and aii rain ondiion o b namd lar. Th apla Tranorm o i dind a an ingral ranorm: F d Th krnl union i K, =. Sin oluion o linar dirnial quaion wih onan oiin ar bad on h xponnial union, h apla ranorm i pariularly uul or uh quaion. No ha h apla Tranorm i dind by an impropr ingral, and hu mu b hkd or onvrgn. On h nx w lid, w rviw xampl o impropr ingral and piwi oninuou union.
Thorm 6.. Suppo ha i a union or whih h ollowing hold: i piwi oninuou on [, b] or all b >. K a whn M, or onan a, K, M, wih K, M >. Thn h apla Tranorm o xi or > a. F d ini No: A union ha aii h ondiion piid abov i aid o o hav xponnial ordr a.
Exampl = or. Thn h apla ranorm F o i: lim b b d lim b, d b
Exampl 5 = a or. Thn h apla ranorm F o i: a lim b b a d a a lim b a, a a d b
Exampl 6 Conidr h ollowing piwi-dind union, k, whr k i a onan. Thi rprn a uni impul. Noing ha i piwi oninuou, w an ompu i apla ranorm { } d, d Obrv ha hi rul do no dpnd on k, h union valu a h poin o dioninuiy.
Exampl 7 = ina or. Uing ingraion by par wi, h apla ranorm F o i ound a ollow: F a a in a F in ad b b lim o a / a b a b lim o a a a b b lim in a / a a a b a F a a lim b b, o a b in a in ad
inariy o h apla Tranorm Suppo and g ar union who apla ranorm xi or > a and > a, rpivly. Thn, or grar han h maximum o a and a, h apla ranorm o + g xi. Tha i, wih ini i d g g g d g d g
Exampl 8 = 5-3in or. Thn by linariy o h apla ranorm, and uing rul o prviou xampl, h apla ranorm F o i:, 6 5 in 3 5 3in 5 } { F
Boy/DiPrima/Mad h d, Ch 6.: Soluion o Iniial Valu Problm Elmnary Dirnial Equaion and Boundary Valu Problm, h diion, by William E. Boy, Rihard C. DiPrima, and Doug Mad 7 by John Wily & Son, In. Th apla ranorm i namd or h Frnh mahmaiian apla, who udid hi ranorm in 78. Th hniqu dribd in hi hapr wr dvlopd primarily by Olivr Haviid 85-95, an Englih lrial nginr. In hi ion w how h apla ranorm an b ud o olv iniial valu problm or linar dirnial quaion wih onan oiin. Th apla ranorm i uul in olving h dirnial quaion bau h ranorm o ' i rlad in a impl way o h ranorm o, a ad in Thorm 6...
Thorm 6.. Suppo ha i a union or whih h ollowing hold: i oninuou and ' i piwi oninuou on [, b] or all b >. K a whn M, or onan a, K, M, wih K, M >. Thn h apla Tranorm o ' xi or > a, wih Proo oulin: For and ' oninuou on [, b], w hav lim b b d lim b lim b b b d d Similarly or ' piwi oninuou on [, b], x. b b b
Th apla Tranorm o ' Thu i and ' aiy h hypoh o Thorm 6.., hn Now uppo ' and '' aiy h ondiion piid or and ' o Thorm 6... W hn obain Similarly, w an driv an xprion or { n }, providd and i drivaiv aiy uiabl ondiion. Thi rul i givn in Corollary 6..
Corollary 6.. Suppo ha i a union or whih h ollowing hold:, ', '',, n- ar oninuou, and n piwi oninuou, on [, b] or all b >. K a, ' K a,, n K a or M, or onan a, K, M, wih K, M >. Thn h apla Tranorm o n xi or > a, wih n n n n n n
Exampl : Chapr 3 Mhod o Conidr h iniial valu problm y y y, y, y Rall rom Sion 3.: r y r r r r Thu r = and r = 3, and gnral oluion ha h orm y y Uing iniial ondiion: Thu y /3 /3 /3, /3 W now olv hi problm uing apla Tranorm. 5 5 y /3 /3..5..5.
y, y y y, y Exampl : apla Tranorm Mhod o Aum ha our IVP ha a oluion and aiy h ondiion o Corollary 6... Thn { y y y} { y } { y} { y} {} and hn { y} y y { y} y { y} ing Y = {y}, w hav Y y y Subiuing in h iniial ondiion, w obain Y Thu { y} Y ' and ''
Exampl : Parial Fraion 3 o Uing parial raion dompoiion, Y an b rwrin: Thu /3, /3, b a b a b a b a b a b a b a } { 3 / 3 / Y y
Exampl : Soluion o Rall rom Sion 6.: Thu Y a a a F d d, a / 3 / 3 /3 Ralling Y = {y}, w hav { } /3 { a }, and hn { y} {/3 /3 } y = 3 + 3 -
Gnral apla Tranorm Mhod Conidr h onan oiin quaion ay Aum ha h oluion y aii h ondiion o Corollary 6.. or n =. W an ak h ranorm o h abov quaion: whr F i h ranorm o. Solving or Y giv: by y a Y- y- y'+ by- y+ Y = F Y = a + by+ ay' a + b + + F a + b +
Algbrai Problm Thu h dirnial quaion ha bn ranormd ino h h algbrai quaion Y a b a y ay b F b or whih w k y = uh ha { } = Y. No ha w do no nd o olv h homognou and nonhomognou quaion paraly, nor do w hav a para p or uing h iniial ondiion o drmin h valu o h oiin in h gnral oluion. a
Chararii Polynomial Uing h apla ranorm, our iniial valu problm ay bom Y by y a b a y ay b y, y, y y F b Th polynomial in h dnominaor i h hararii polynomial aoiad wih h dirnial quaion. Th parial raion xpanion o Y ud o drmin rquir u o ind h roo o h hararii quaion. For highr ordr quaion, hi may b diiul, pially i h roo ar irraional or omplx. a
Invr Problm Th main diiuly in uing h apla ranorm mhod i drmining h union y = uh ha { } = Y. Thi i an invr problm, in whih w ry o ind uh ha = {Y}. Thr i a gnral ormula or, bu i rquir knowldg o h hory o union o a omplx variabl, and w do no onidr i hr. I an b hown ha i i oninuou wih {} = F, hn i h uniqu oninuou union wih = {F}. Tabl 6.. in h x li many o h union and hir ranorm ha ar nounrd in hi hapr.
inariy o h Invr Tranorm Frqunly a apla ranorm F an b xprd a Thn h union ha h apla ranorm F, in i linar. By h uniqun rul o h prviou lid, no ohr oninuou union ha h am ranorm F. Thu i a linar opraor wih F F F F n,, F F n n n F F F n
Exampl : Nonhomognou Problm o Conidr h iniial valu problm y y in, y, y Taking h apla ranorm o h dirnial quaion, and auming h ondiion o Corollary 6.. ar m, w hav ing Y = {y}, w hav Subiuing in h iniial ondiion, w obain Y / Thu { y} y y { y} / Y y y / 3 8 6 Y
Uing parial raion, Thn Solving, w obain A =, B = 5/3, C =, and D = -/3. Thu Hn Exampl : Soluion o 6 8 3 D C B A Y 6 8 3 3 D B C A D B C A D C B A / 3 5/ 3 Y y in 3 in 3 5 o
Exampl 3: Solving a h Ordr IVP o Conidr h iniial valu problm y y, y,, y'', y''' Taking h apla ranorm o h dirnial quaion, and auming h ondiion o Corollary 6.. ar m, w hav ing Y = {y} and ubiuing h iniial valu, w hav Y Uing parial raion a b d Y Thu a b d y' 3 { y} y y y'' y''' { y}
y y, y, y', y'', y''' Exampl 3: Solving a h Ordr IVP o In h xprion: Sing = and = nabl u o olv or a and b: a b and a b a, b / Sing =, b d =, o d = / Equaing h oiin o 3in h ir xprion giv a + =, o = Thu / / Y Uing Tabl 6.., h oluion i y inh in a b d y 5 5 inh in y 3 5 6 7
Boy/DiPrima/Mad h d, Ch 6.3: Sp Funion Elmnary Dirnial Equaion and Boundary Valu Problm, h diion, by William E. Boy, Rihard C. DiPrima, and Doug Mad 7 by John Wily & Son, In. Som o h mo inring lmnary appliaion o h apla Tranorm mhod our in h oluion o linar quaion wih dioninuou or impuliv oring union. In hi ion, w will aum ha all union onidrd ar piwi oninuou and o xponnial ordr, o ha hir apla Tranorm all xi, or larg nough.
Sp Funion diniion >. Th uni p union, or Haviid union, i dind by A ngaiv p an b rprnd by u,, u y,,
Exampl Skh h graph o y = h, whr Soluion: Rall ha u i dind by Thu and hn h graph o h i a rangular pul., u u h u,, h,,
Exampl For h union, 5, h,, who graph i hown To wri h in rm o u, w will nd u, u 7, and u 9. W bgin wih h, hn add 3 o g 5, hn ubra 6 o g, and inally add o g ah quaniy i muliplid by h appropria u 7 9 7 9 h 3u 6u7 u9,
apla Tranorm o Sp Funion Th apla Tranorm o u i u lim b b b lim u d b d b lim d b
Tranlad Funion Givn a union dind or, w will on wan o onidr h rlad union g = u - : g,, Thu g rprn a ranlaion o a dian in h poiiv dirion. In h igur blow, h graph o i givn on h l, and h graph o g on h righ.
Thorm 6.3. I F = { } xi or > a, and i >, hn Convrly, i = {F}, hn Thu h ranlaion o a dian in h poiiv dirion orrpond o a mulipliaion o F by. F u F u
Thorm 6.3.: Proo Oulin W nd o how Uing h diniion o h apla Tranorm, w hav F du u du u d d u u u u u F u
Exampl 3 Find { }, whr i dind by No ha = in + u / o /, and = in, < p in + o - p, ³ p ì í ï ï î ï ï o in / o in / / / / u p p
Exampl Find {F}, whr Soluion: Th union may alo b wrin a F u,,
Thorm 6.3. I F = { } xi or > a, and i i a onan, hn Convrly, i = - {F}, hn Thu mulipliaion by rul in ranlaing F a dian in h poiiv dirion, and onvrly. Proo Oulin: a F, F F d d
Exampl 5 To ind h invr ranorm o W ir ompl h quar: Sin i ollow ha 5 G 5 F G G g o and o F F
Boy/DiPrima/Mad h d, Ch 6.: Dirnial Equaion wih Dioninuou Foring Funion Elmnary Dirnial Equaion and Boundary Valu Problm, h diion, by William E. Boy, Rihard C. DiPrima, and Doug Mad 7 by John Wily & Son, In. In hi ion ou on xampl o nonhomognou iniial valu problm in whih h oring union i dioninuou. ay by y y, y g, y y
Exampl : Iniial Valu Problm o Find h oluion o h iniial valu problm whr y y y g, y y, 5 g u5 u, 5 and Suh an iniial valu problm migh modl h rpon o a dampd oillaor ubj o g, or urrn in a irui or a uni volag pul.,
Exampl : apla Tranorm o Aum h ondiion o Corollary 6.. ar m. Thn or ing Y = {y}, Subiuing in h iniial ondiion, w obain Thu } { } { } { } { } { 5 u u y y y y y y y y y 5 } { } { } { y y Y 5 Y 5 5 Y,, 5 y y u u y y y
Exampl : Faoring Y 3 o W hav whr I w l h = - {H}, hn by Thorm 6.3.. 5 5 h u h u y 5 5 H Y H
Exampl : Parial Fraion o Thu w xamin H, a ollow. A B C H Thi parial raion xpanion yild h quaion A B A C A A /, B, C / Thu H / /
Exampl : Compling h Squar 5 o Compling h quar, 5/6 / / / / 5/6 / / / 5 /6 /6 / / / / / / / / H
Exampl : Soluion 6 o Thu and hn For h a givn abov, and ralling our prviou rul, h oluion o h iniial valu problm i hn 5/6 / 5 / 5 5/6 / / / 5/6 / / / / H H h 5 in 5 5 o } { / / 5 5 h u h u
Exampl : Soluion Graph 7 o Thu h oluion o h iniial valu problm i h u 5 h 5 u o h, 5 in 5 / / 5 Th graph o hi oluion i givn blow. whr
Exampl : Compoi IVP 8 o Th oluion o original IVP an b viwd a a ompoi o hr para oluion o hr para IVP: 5 : 5: : y y y 3 y y y y 3 y y 3,,, y y y, 3 5, y y y 5, y 3 y
Exampl : Fir IVP 9 o Conidr h ir iniial valu problm y y y, y, y ; 5 From a phyial poin o viw, h ym i iniially a r, and in hr i no xrnal oring, i rmain a r. Thu h oluion ovr [, 5 i y =, and hi an b vriid analyially a wll. S graph blow.
Exampl : Sond IVP o Conidr h ond iniial valu problm y y y, y5, y 5 ; 5 Uing mhod o Chapr 3, h oluion ha h orm / 5 / in 5 / / / y o Phyially, h ym rpond wih h um o a onan h rpon o h onan oring union and a dampd oillaion, ovr h im inrval 5,. S graph blow.
Exampl : Third IVP o Conidr h hird iniial valu problm y 3 y 3 y3, y3 y, y 3 y ; Uing mhod o Chapr 3, h oluion ha h orm 5 / in 5 / / / y3 o Phyially, in hr i no xrnal oring, h rpon i a dampd oillaion abou y =, or >. S graph blow.
Exampl : Soluion Smoohn o Our oluion i u I an b hown ha and ar oninuou a = 5 and =, and ha a jump o / a = 5 and a jump o / a = : 5 h 5 u h '' limj =, 5 - ' limj = / 5 + lim j @.7, lim j @.57 - + Thu jump in oring rm g a h poin i baland by a orrponding jump in high ordr rm y'' in ODE.
Smoohn o Soluion in Gnral Conidr a gnral ond ordr linar quaion y whr p and q ar oninuou on om inrval a, b bu g i only piwi oninuou hr. I y = i a oluion, hn and ar oninuou on a, b bu poin a g. p y q y g ' '' ha jump dioninuii a h am Similarly or highr ordr quaion, whr h high drivaiv o h oluion ha jump dioninuii a h am poin a h oring union, bu h oluion il and i lowr drivaiv ar oninuou ovr a, b.
Exampl : Iniial Valu Problm o Find h oluion o h iniial valu problm whr y + y = g, y =, y = g = u 5-5 5 - u - 5 Th graph o oring union g i givn on righ, and i known a ramp loading. ì ï = í ï î ï, < 5-5 5 5 <, ³
y 5 u5 u, y, y 5 5 y Exampl : apla Tranorm o Aum ha hi ODE ha a oluion y = and ha ' '' and aiy h ondiion o Corollary 6... Thn { y } { y} { u5 5 } 5{ u } 5 or 5 5 { y} y y { y} ing Y = {y}, and ubiuing in iniial ondiion, Thu 5 Y 5 5 Y 5
Exampl : Faoring Y 3 o W hav whr I w l h = - {H}, hn by Thorm 6.3.. 5 5 5 h u h u y 5 5 5 5 H Y H
Exampl : Parial Fraion o Thu w xamin H, a ollow. A B C H Thi parial raion xpanion yild h quaion D A C 3 B D A, B /, A B C, D / Thu H / /
Exampl : Soluion 5 o Thu and hn For h a givn abov, and ralling our prviou rul, h oluion o h iniial valu problm i hn H h in 8 } { 8 / / H 5 5 5 h u h u y
Exampl : Graph o Soluion 6 o Thu h oluion o h iniial valu problm i Th graph o hi oluion i givn blow. h h u h u in 8 whr, 5 5 5
Exampl : Compoi IVP 7 o Th oluion o original IVP an b viwd a a ompoi o hr para oluion o hr para IVP diu: 5: y y, y, y 5 : : y y 3 y y 3, 5 / 5, y y 3 5, y y 5, y 3 y
Exampl : Fir IVP 8 o Conidr h ir iniial valu problm y y, y, y ; 5 From a phyial poin o viw, h ym i iniially a r, and in hr i no xrnal oring, i rmain a r. Thu h oluion ovr [, 5 i y =, and hi an b vriid analyially a wll. S graph blow.
Exampl : Sond IVP 9 o Conidr h ond iniial valu problm y y 5 /5, y5, y 5 ; 5 Uing mhod o Chapr 3, h oluion ha h orm in / / y o Thu h oluion i an oillaion abou h lin 5/, ovr h im inrval 5,. S graph blow.
Exampl : Third IVP o Conidr h hird iniial valu problm y 3 y3, y3 y, y 3 y ; Uing mhod o Chapr 3, h oluion ha h orm in / y3 o Thu h oluion i an oillaion abou y = /, or >. S graph blow.
Exampl : Ampliud o Rall ha h oluion o h iniial valu problm i y 5 5 8 u h 5 u h, h in To ind h ampliud o h vnual ady oillaion, w loa on o h maximum or minimum poin or >. Solving y' =, h ir maximum i.6,.979. Thu h ampliud o h oillaion i abou.79.
Exampl : Soluion Smoohn o Our oluion i y 5 5 8 u h 5 u h, h in In hi xampl, h oring union g i oninuou bu g' i dioninuou a = 5 and =. ''' I ollow ha and i ir wo drivaiv ar oninuou vrywhr, bu ha dioninuii a = 5 and = ha mah h dioninuii o g' a = 5 and =.
Boy/DiPrima/Mad h d, Ch 6.5: Impul Funion Elmnary Dirnial Equaion and Boundary Valu Problm, h diion, by William E. Boy, Rihard C. DiPrima, and Doug Mad 7 by John Wily & Son, In. In om appliaion, i i nary o dal wih phnomna o an impuliv naur. For xampl, an lrial irui or mhanial ym ubj o a uddn volag or or g o larg magniud ha a ovr a hor im inrval abou. Th dirnial quaion will hn hav h orm whr ay by y g, big, g, ohrwi and i mall.
Mauring Impul In a mhanial ym, whr g i a or, h oal impul o hi or i maurd by h ingral No ha i g ha h orm hn In pariular, i = /, hn I = indpndn o. d g d g I ohrwi,, g, d g d g I
Uni Impul Funion Suppo h oring union ha h orm Thn a w hav n, I =. W ar inrd aing ovr horr and horr im inrval i..,. S graph on righ. No ha g allr and narrowr a lim d d = ì ï í ï î d, - < <, ohrwi. Thu or, w hav, and d lim I d
Thu or, w hav Dira Dla Funion lim d and lim I Th uni impul union i dind o hav h propri or, and d Th uni impul union i an xampl o a gnralizd union and i uually alld h Dira dla union. In gnral, or a uni impul a an arbirary poin, d, or, and d
apla Tranorm o o Th apla Tranorm o i dind by and hu, lim d oh lim inh lim lim lim lim lim lim d d d d d
apla Tranorm o o d Thu h apla Tranorm o i, For apla Tranorm o a =, ak limi a ollow: lim d lim d For xampl, whn =, w hav { } =. d d
Produ o Coninuou Funion and Th produ o h dla union and a oninuou union an b ingrad, uing h man valu horm or ingral: Thu * lim * whr * lim lim lim d d d d d d
Exampl : Iniial Valu Problm o 3 Conidr h oluion o h iniial valu problm y y y 5, y, y Thn { y } { y} { y} { 5} ing Y = {y}, 5 Y y y Y y Y Subiuing in h iniial ondiion, w obain or 5 Y 5 Y 5
Exampl : Soluion o 3 W hav 5 Y Th parial raion xpanion o Y yild and hn 5 5 / Y 5 / 5/6 y.5. Plo o h Soluion y 5 u in 5 5/ 5 5.3... 5 5.
Exampl : Soluion Bhavior 3 o 3 Wih homognou iniial ondiion a = and no xrnal xiaion unil = 5, hr i no rpon on, 5. Th impul a = 5 produ a daying oillaion ha pri indinily. Rpon i oninuou a = 5 dpi ingulariy in oring union. Sin y' ha a jump dioninuiy a = 5, y'' ha an inini dioninuiy hr. Thu a ingulariy in h oring union i baland by a orrponding ingulariy in y''. 5 y.5..3 Plo o h Soluion.. 5 5..
Boy/DiPrima/Mad h d, Ch 6.6: Th Convoluion Ingral Elmnary Dirnial Equaion and Boundary Valu Problm, h diion, by William E. Boy, Rihard C. DiPrima, and Doug Mad 7 by John Wily & Son, In. Somim i i poibl o wri a apla ranorm H a H = FG, whr F and G ar h ranorm o known union and g, rpivly. In hi a w migh xp H o b h ranorm o h produ o and g. Tha i, do H = FG = { }{g} = { g}? On h nx lid w giv an xampl ha how ha hi qualiy do no hold, and hn h apla ranorm anno in gnral b ommud wih ordinary mulipliaion. In hi ion w xamin h onvoluion o and g, whih an b viwd a a gnralizd produ, and on or whih h apla ranorm do ommu.
Obrvaion = and g = in. Rall ha h apla Tranorm o and g ar Thu and Thror or h union i ollow ha in g in, g g g g
Thorm 6.6. Suppo F = { } and G = {g} boh xi or > a. Thn H = FG = {h} or > a, whr h g d g d Th union h i known a h onvoluion o and g and h ingral abov ar known a onvoluion ingral. No ha h qualiy o h wo onvoluion ingral an b n by making h ubiuion u = x. Th onvoluion ingral din a gnralizd produ and an b wrin a h = *g. S x or mor dail.
Thorm 6.6. Proo Oulin h d d g d d g dd g u d d g du u d g d g du u G F u u
Exampl : Find Invr Tranorm o Find h invr apla Tranorm o H, givn blow. Soluion: F = / and G = a/ + a, wih Thu by Thorm 6.6., in a G g F a a H d a h H in
Exampl : Soluion h o W an ingra o impliy h, a ollow. in in in ] [o o o o o in in in a a a a a a a a a a a a d a a a a a a d a d a d a h d a h H in
Exampl : Iniial Valu Problm o Find h oluion o h iniial valu problm Soluion: or ing Y = {y}, and ubiuing in iniial ondiion, Thu } { } { } { g y y } { } { G y y y y 3 G Y 3 G Y 3,, y y g y y
Exampl : Soluion o W hav Thu No ha i g i givn, hn h onvoluion ingral an b valuad. d g y in in 3o 3 3 G G Y
Exampl : apla Tranorm o Soluion 3 o Rall ha h apla Tranorm o h oluion y i 3 G Y Φ Ψ y y g, y 3, y No F dpnd only on ym oiin and iniial ondiion, whil Y dpnd only on ym oiin and oring union g. Furhr, = [ F ] olv h homognou IVP y y, y 3, y whil y = { } olv h nonhomognou IVP Y y y g, y, y
Exampl : Tranr Funion o Examining mor loly, Y G Ψ H G, whr H Th union H i known a h ranr union, and dpnd only on ym oiin. Th union G dpnd only on xrnal xiaion g applid o ym. I G =, hn g = d and hn h = {H} olv h nonhomognou iniial valu problm y y, y, y Thu h i rpon o ym o uni impul applid a =, and hn h i alld h impul rpon o ym.