Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors

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Boc/DiPrima 9 h d, Ch.: 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. Exampls 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 nx 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.

Exampl : Ingraing Facor of Considr h following quaion: /3 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 [ ] / /3

Exampl : Gnral Soluion of Wih /, w solv h original quaion as follows: [ ] gnral soluion : / /3 / 6 5 / / 6 5 / / 6 5 / / /3 5 3 5 3 C C d d d d 3 4 5 6-3 HL / /3 5 3 Sampl Soluions : C

Mhod of Ingraing Facors: Variabl Righ Sid In gnral, for variabl righ sid g, h soluion can b found as follows: a g a d d a d d [ ] a d d a a a g a a a a g g d a g g d C a

Exampl : Gnral Soluion of W can solv h following quaion using h formula drivd on h prvious slid: Ingraing b pars, Thus 4 a a a g d C 4 d 7 4 4 d 4 7 4 /5 C 7 4 d C C d d

4 Exampl : Graphs of Soluions of Th graph shows h dircion fild along wih svral ingral curvs. If w s C 0, h xponnial rm drops ou and ou should noic how h soluion in ha cas, hrough h poin 0, -7/4, sparas h soluions ino hos ha grow xponniall in h posiiv dircion from hos ha grow xponniall in h ngaiv dircion.. 7 4 0 C 0.5.0.5.0 - HL - -3-4

Mhod of Ingraing Facors for Gnral Firs Ordr Linar Equaion Nx, w considr h gnral firs ordr linar quaion p g Mulipling boh sids b, w obain d p g d Nx, 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,

Exampl 3: Gnral Soluion of To solv h iniial valu problm firs pu ino sandard form: Thn and hnc 4, 4, for, 0 p d d ln ln g d C 4 d C [ ] 3 C 4 d C

Exampl 3: 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 and ha soluion sparas h soluions ino hos ha ar asmpoic o h posiiv vrsus ngaiv -axis. 4, 4 3, - - - - 5, C

Exampl 4: 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 4 ds C

Exampl 4: 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 Using sofwar such as Mahmaica or Mapl, w can approxima h soluion for h givn iniial condiions as HL wll as for ohr iniial 3 condiions. Svral soluion curvs ar shown. 3 4 5 6 ds C - - -3 0,, 0 s ds C

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