3.4 Repeated Roots; Reduction of Order

Transcription

1 3.4 Rpd Roos; Rducion of Ordr Rcll our nd ordr linr homognous ODE b c 0 whr, b nd c r consns. Assuming n xponnil soluion lds o chrcrisic quion: r r br c 0 Qudric formul or fcoring ilds wo soluions, r & r : r b b 4c Whn b 4c = 0, r = r = -b, sinc mhod onl gis on soluion: c b

2 Exmpl ODE: Find gnrl soluion of h '' ' 0

3 Exmpl Find gnrl soluion of h ODE: '' ' 0 r r r 0 r :?? Qusions: Is i rll soluion of h ODE? Ar h wo soluions fundmnl soluions? How do w compu?

4 Scond Soluion: Mulipling Fcor W know h Sinc nd r linrl dpndn, w gnrliz his pproch nd mulipl b funcion, nd drmin condiions for which is soluion: Thn soluion soluion c soluion r b b b b b b b b b b b b b 4

5 Finding Mulipling Fcor b c 0 Subsiuing driis ino ODE, w sk formul for : b b b b b c 0 4 b b b b c 0 4 b b c 0 4 b b 4c b 4c b 4c k k 3 4

6 Gnrl Soluion To find our gnrl soluion, w h: k k b k c b b c k 3 k b b 4 b Thus h gnrl soluion for rpd roos is c c b b

7 Wronskin Th gnrl soluion is c b c b Thus r soluion is linr combinion of, b b Th Wronskin of h wo soluions is W, b b b b b b b b 0 b for ll b b Thus nd form fundmnl soluion s for quion.

8 Exmpl Find gnrl soluion of h ODE. '' ' ' 4 4 0, 0, 0 3

9 Exmpl of Considr h iniil lu problm Assuming xponnil soluion lds o chrcrisic quion: So on soluion is nd scond soluion is found: Subsiuing hs ino h diffrnil quion nd simplifing ilds whr r rbirr consns r r r r r 4 4, ' 0, " k k k nd c c

10 Exmpl of Ling k nd k 0, n d So h gnrl soluion is c c No h boh nd o 0 s rgrdlss of h lus of c nd c Using iniil condiions nd 0 nd c c c '0 3 c 3, c Thrfor h soluion o h IVP is

11 Exmpl 3 Find h soluion of h IVP '' ' ' 0.5 0, 0, 0 3

12 Exmpl 3 of Considr h iniil lu problm Assuming xponnil soluion lds o chrcrisic quion: Thus h gnrl soluion is Using h iniil condiions: Thus 3 0, 0 0, r r r r r c c 3, 3 c c c c c

13 Exmpl 3 of Suppos h h iniil slop in h prious problm ws incrsd 0, 0 Th soluion of his modifid problm is Noic h h cofficin of h scond rm is now posii. This mks big diffrnc in h grph, sinc h xponnil funcion is risd o posii powr: rd: blu : 3 3 4

14 Eulr quions: '' ' b c 0 Considr scond ordr DE wih h following ribl cofficins: A B C '' ' 0 A, B b, C c Qusion Wh kind of funcions cn b is soluions? Exponnil funcion? Wh is h fur of h ODE? How do w find is gnrl soluion?

15 Exmpl Eulr quions '' ' 0 '' ' '' ' 3 0

16 Rducion of Ordr Th mhod usd so fr in his scion lso works for quions wih nonconsn cofficins: Th is, gin h is soluion, r = : Subsiuing hs ino ODE nd collcing rms, Sinc is soluion o h diffrnil quion, his ls quion rducs o firs ordr quion in : 0 q p 0 q p p 0 p

17 Exmpl 4: Rducion of Ordr of 3 Gin h ribl cofficin quion Eulr quion nd soluion, 3 0, 0;, us rducion of ordr mhod o find scond soluion: Subsiuing hs ino h ODE nd collcing rms, u u 0, whr u

18 Exmpl 4: Finding of 3 To sol u u 0, u for u, w cn us h sprion of ribls mhod: du du u 0 d ln u ln C d u C u u c, sinc 0. Thus c nd hnc c ln k

19 Exmpl 4: Gnrl Soluion 3 of 3 Sinc c ln k, c ln k c ln k Rcll So w cn nglc h scond rm of o obin ln Th Wronskin of nd cn b compud W, 3 3 0, 0 Hnc h gnrl soluion o h diffrnil quion is c c ln

20 Quiz Find h soluion of h iniil lu problm '' ' ' 5 0, 0 0, 0 4