Ordinary Differential Equations

Transcription

1 Basi Nomlatur MAE 0 all 005 Egirig Aalsis Ltur Nots o: Ordiar Diffrtial Equatios Author: Profssor Albrt Y. Tog Tpist: Sakurako Takahashi Cosidr a gral O. D. E. with t as th idpdt variabl, ad th dpdt variabl. d d a0, a, a, a, f dt dt dt If f 0, th th quatio is homogous. f 0, th th quatio is o-homogous. Th ordr of th D. E. is, th ordr of th highst drivativ. Th quatio is liar if a 0, a,, a all dpd ol o t, th idpdt variabl, or ar ostats. Th quatio is ordiar if it otais ol ordiar drivativs. A solutio of a O. D. E. is a futioal rlatio that satisfis th D. E. i a form ot otaiig itgrals or drivativs of ukow futios. Solutio of first ordr quatios or i th gral form i.. Equatio, th homogous quatio is: a 0, a, 0 dt Rarragig & ramig thigs; it boms d 0. Eat diffrtial

2 Cosidr, Takig total drivativs 0 d Comparig with Equatio givs, ad, But This is a tst to s if th D. E. is a at diffrtial. If it is, th w a itgrat ad as show i th ampl blow: Eampl id a solutio to: ad it is a at diffrtial, ad. 0 d, givs : f f f d

3 givs : l Comparig ad givs: f l f ad l f f f or l is th solutio.. s of a itgratig fator If d is NOT a at diffrtial, somtims it a b mad o b multiplig b a itgratig fator. i.. p d p ma b a at diffrtial. Eampl Cosidr d 0 Not : d d d If w multipl b th itgratig fator d d 0 solutio :, th th rsult is a at diffrtial.

4 Cosidr th stadard form of liar D.E. of first ordr blow: Not that th offiit of has b ormalizd to d a f d To solv: multipl b a itgratig fator p p a p p f d Compar with: d dp [ p ] p d d d Lft-had sid of will b a at diffrtial if: dp p a d dp a d p l p or p a d a d With p dtrmid this wa, from p p a d d p p f d [ p ] p f p f d p p o-homogous solutio homogous solutio or a d a d a d f d

5 5 Eampls o Itgratig ators Eampl # d d d d d p d Eampl # Not : u u d du d d d d d d p d

6 Eampl # d d d d p d d Eampl # d d d d p a d d d a si os s os si si si si os si si si ot si si ot os ot os lsi ot

7 Broulli's Equatio p Q ; ostat d Not that it is a oliar D. E. But it is rduibl to liar form b a hag of variabl. Also ot that th D. E. is giv i a diffrt form. Suppos that. othrwis th quatio a b rarragd as [ P Q ] 0 This a b solvd b rparatio of variabl. Dividig b Lt v d givs: P d d Equatio boms: Q Chag of Variabl P v Q d Th prvious thiqu usig itgratig fator a b applid. d Eampl: Solv l d l d Q P Lt v d d Equatio boms: 7

8 l d l d or v l d Q P Itgratig ator: P d d l l l v l d d v d v l d Not l d l. Sparatio of variabl d 0 If th futios ad ar fatorabl; i.. th th solutio is simpl: d 0 divid b : d 0 d 8

9 d This quatio ma b writt i th form Eampl : Solv th D. E. 0 d whr, i itgratio, ta si 0 Eampl : d 0 This ma b writt i th form: l l d 0 d l l [ ] l l. Chag of variabls Som diffrtial quatios a b rdud ito simplr form b a hag of variabl. Of partiular valu would b a hag of variabl that rdus a oliar quatio to a liar o. O tp of quatio whr this thiqu is sussful is wh ad ar polomial of sam dgr or similar futio. Dfiitio: A futio is homogous to dgr if thr is a ostat suh that for vr λ 9

10 λ λ λ.g. 5 If i d 0, ad ar both homogous to dgr, th th substitutio, quatio. Lt v ; vd th D. E. boms: v or u hoos o ovi, lads to a sparabl [ vd ] 0 v d v du but v, v ad v, v baus ad ar homogous to dgr [Not: th λ is dfid as i this as] [ vd ], v d, v 0 [, v v, v ] d, v d, v, v v, v, v l, v v, v ampl Solv d 0 B isptio ad ar homogous to, lt v ; vd th: v v d vd v v d vd v d d v l v or with v; l 0 0 0