Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ กด วล ยร ชต
Topics Linar Equaions; Mhod of Ingraing Facors Sparabl Equaions Eac Equaions and Ingraion Facors Numrical Approimaions: Eulr s Mhod Th Eisnc and Uniqunss Thorm Firs Ordr Diffrnc Equaions
.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms 9 h diion b William E. Boc and Richard C. DiPrima 009 b John Wil & Sons Inc. A linar firs ordr ODE has h gnral form d d f whr f is linar in. Eampls includ quaions wih consan cofficins such as hos in Chapr a b or quaions wih variabl cofficins: d d p g
Consan Cofficin Cas For a firs ordr linar quaion wih consan cofficins a b rcall ha w can us mhods of calculus o solv: d / d a b / a d b / a ln b / a a C b / a k a a d k ± C
Variabl Cofficin Cas: Mhod of Ingraing Facors W n considr linar firs ordr ODEs wih variabl cofficins: d d p g Th mhod of ingraing facors involvs mulipling his quaion b a funcion chosn so ha h rsuling quaion is asil ingrad. 5
Eampl : Ingraing Facor of Considr h following quaion: / Mulipling boh sids b w obain W will choos so ha lf sid is drivaiv of known quani. Considr h following and rcall produc rul: Choos so ha d d d d d d d d [ ] / / 6
Eampl : Gnral Soluion of Wih / w solv h original quaion as follows: [ ] soluion : gnral / / / 6 5 / / 6 5 / / 6 5 / / / 5 5 C C d d d d 5 6 / / 5 Sampl Soluions : C 7
Mhod of Ingraing Facors: Variabl Righ Sid In gnral for variabl righ sid g h soluion can b found as follows: a g d a g d a d a a a g d d [ ] a a g d a a g d a a g d C a 8
Eampl : Gnral Soluion of W can solv h following quaion using h formula drivd on h prvious slid: a a a g d C d C Ingraing b pars Thus 7 d 7 /5 C 7 d C d 9 d
Eampl : Graphs of Soluions of Th graph shows h dircion fild along wih svral ingral curvs. If w s C 0 h ponnial rm drops ou and ou should noic how h soluion in ha cas hrough h poin 0-7/ sparas h soluions ino hos ha grow ponniall in h posiiv dircion from hos ha grow ponniall in h ngaiv dircion.. 7 C 0 0.5.0.5.0 0
Mhod of Ingraing Facors for Gnral Firs Ordr Linar Equaion N w considr h gnral firs ordr linar quaion p g Mulipling boh sids b w obain d p g d N w wan such ha ' p from which i will follow ha d d d d [ ] p
Ingraing Facor for Gnral Firs Ordr Linar Equaion Thus w wan o choos such ha ' p. Assuming > 0 i follows ha d p d p d ln k Choosing k 0 w hn hav p d and no > 0 as dsird.
Soluion for Gnral Firs Ordr Linar Equaion Thus w hav h following: Thn d p g p d d g p whr [ ] d p c d g c d g g d d whr
Eampl : Gnral Soluion of To solv h iniial valu problm firs pu ino sandard form: Thn and hnc for 0 p d d ln ln g d C d C [ ] C d C
Eampl : Paricular Soluion of Using h iniial condiion and gnral soluion C C C i follows ha Th graphs blow show soluion curvs for h diffrnial quaion including a paricular soluion whos graph conains h iniial poin. Noic ha whn C0 w g h parabolic soluion shown 5 and ha soluion sparas h soluions ino hos ha ar asmpoic o h posiiv vrsus C ngaiv -ais. 5
Eampl : A Soluion in Ingral Form of To solv h iniial valu problm firs pu ino sandard form: Thn and hnc 0 d p d / s / / ds C 0 0 s / ds C / 6
Eampl : A Soluion in Ingral Form of Noic ha his soluion mus b lf in h form of an ingral sinc hr is no closd form for h ingral. / 0 s / ds C Using sofwar such as Mahmaica or Mapl w can approima h soluion for h givn iniial condiions as wll as for ohr iniial condiions. Svral soluion curvs ar shown. 5 6 / 0 / 0 s / ds C / 7
.: Sparabl Equaions Elmnar Diffrnial Equaions and Boundar Valu Problms 9 h diion b William E. Boc and Richard C. DiPrima 009 b John Wil & Sons Inc. In his scion w amin a subclass of linar and nonlinar firs ordr quaions. Considr h firs ordr quaion d d f W can rwri his in h form d M N 0 d For ampl l M - f and N. Thr ma b ohr was as wll. In diffrnial form M d N d If M is a funcion of onl and N is a funcion of onl hn M d N d In his cas h quaion is calld sparabl. 0 0 8
Eampl : Solving a Sparabl Equaion Solv h following firs ordr nonlinar quaion: d d Sparaing variabls and using calculus w obain d d d C C d Th quaion abov dfins h soluion implicil. A graph showing h dircion fild and implici plos of svral soluion curvs for h diffrnial quaion is givn abov. 9
Eampl : Implici and Eplici Soluions of Solv h following firs ordr nonlinar quaion: Sparaing variabls and using calculus w obain Th quaion abov dfins h soluion implicil. An plici prssion for h soluion can b found in his cas: d d C d d d d C C C ± ± 0 0
Eampl : Iniial Valu Problm of Suppos w sk a soluion saisfing 0 -. Using h implici prssion of w obain Thus h implici quaion dfining is Using an plici prssion of I follows ha ± ± C C C C C C d d
d d Eampl : Iniial Condiion 0 of No ha if iniial condiion is 0 hn w choos h posiiv sign insad of ngaiv sign on h squar roo rm:
Eampl : Domain of Thus h soluions o h iniial valu problm d 0 d ar givn b implici plici From plici rprsnaion of i follows ha and hnc h domain of is -. No - ilds which maks h dnominaor of d/d zro vrical angn. Convrsl h domain of can b simad b locaing vrical angns on h graph usful for implicil dfind soluions.
Eampl : Implici Soluion of an Iniial Valu Problm of Considr h following iniial valu problm: 0 Sparaing variabls and using calculus w obain d d 6 8 d C d c whr C c Using h iniial condiion 0 i follows ha C 7. 6 8 7
Eampl : Graph of Soluions of Thus h gnral soluion is 6 8 and h soluion hrough 0 is 6 8 7 Th graph of his paricular soluion hrough 0 is shown in rd along wih h graphs of h dircion fild and svral ohr soluion curvs for his diffrnial quaion ar shown: Th poins idnifid wih blu dos corrspond o h poins on h rd curv whr h angn lin is vrical: -.587 ±.88 on h rd curv bu a all poins whr h lin conncing h blu poins inrscs soluion curvs h angn lin is vrical. 0 C 5
.6: Eac Equaions & Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms 9 h diion b William E. Boc and Richard C. DiPrima 009 b John Wil & Sons Inc. Considr a firs ordr ODE of h form M N 0 Suppos hr is a funcion ψ such ha ψ M ψ N and such ha ψ c dfins φ implicil. Thn ψ ψ d d M N ψ d d and hnc h original ODE bcoms d ψ d [ φ ] 0 Thus ψ c dfins a soluion implicil. In his cas h ODE is said o b ac. [ φ ] 6
Eampl : Eac Equaion Considr h quaion: I is nihr linar nor sparabl bu hr is a funcion ψ such ha ψ ψ and Th funcion ha works is ψ Thinking of as a funcion of and calling upon h chain rul h diffrnial quaion and is soluion bcom dψ d d d ψ 0 0 c d c d 7
Thorm.6. Suppos an ODE can b wrin in h form M N whr h funcions M N M and N ar all coninuous in h rcangular rgion R: α β γ δ. Thn Eq. is an ac diffrnial quaion iff M N R Tha is hr iss a funcion ψ saisfing h condiions iff M and N saisf Equaion. 0 ψ M ψ N 8
Eampl : Eac Equaion of Considr h following diffrnial quaion. Thn and hnc From Thorm.6. 0 sin cos sin cos N M ODE is ac cos N M sin cos N M ψ ψ 9
Eampl : Soluion of W hav and I follows ha Thus B Thorm.6. h soluion is givn implicil b sin cos N M ψ ψ sin cos C d d ψ ψ k C C C sin sin ψ k sin ψ c sin 0
Eampl : Dircion Fild and Soluion Curvs of Our diffrnial quaion and soluions ar givn b cos sin 0 sin c A graph of h dircion fild for his diffrnial quaion along wih svral soluion curvs is givn blow.
Eampl : Non-Eac Equaion of Considr h following diffrnial quaion. Thn and hnc To show ha our diffrnial quaion canno b solvd b his mhod l us sk a funcion ψ such ha ψ ψ Thus M 0 M N N M N ODE is no ac d / C ψ d ψ
Eampl : Non-Eac Equaion of W sk ψ such ha and Thn ψ M ψ N d / C ψ ψ d ψ? C / / C Bcaus C dpnds on as wll as hr is no such funcion ψ such ha dψ d
I is somims possibl o convr a diffrnial quaion ha is no ac ino an ac quaion b mulipling h quaion b a suiabl ingraing facor : For his quaion o b ac w nd This parial diffrnial quaion ma b difficul o solv. If is a funcion of alon hn 0 and hnc w solv providd righ sid is a funcion of onl. Similarl if is a funcion of alon. S for mor dails. Ingraing Facors 0 0 N M N M 0 N M N M N M N N M d d
Eampl : Non-Eac Equaion Considr h following non-ac diffrnial quaion. Sking an ingraing facor w solv h linar quaion d M N d d N d Mulipling our diffrnial quaion b w obain h ac quaion which has is soluions givn implicil b 0 c 0 5