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ODE PDE Wat is te differece ordiar differetial eqatio partial differetial eqatio? 94. order 4. degree 7 d d d d d d 4 e e d d 4 cos 5. liear a a L a r 立數 c~6 -

7 d d d d d d 4 e e d d 4 cos 6. soltio geeral soltio particlar soltio siglar soltio P4-6 a Sow tat c c geeral soltio of. b Fid particlar soltio ad 5.95 P-79 7 siτ dτ τ is te soltio of ' si,, %95 4 " cos ' 5si as a soltio satisfig,, ad C if ad ol if C.%95 立數 c~6 -

5 ad be particlar soltios, cos e e cos, ' cos e e si, ' Te is a particlar soltio cos e e cos si,, ' %95 e, e m m e e d d e d d e e m e m f f e e e f l 立數 c~6 -

l l l l >, > l l l >, > 4 l l > d l d d f l f d f l e e l 立數 c~6-4

sit 4 5 6 liear - d f, f f d d 數 c f d c f d f d c c f c d f f f d f 數了 f df d l f c f f l d l c 5 立數 c~6-5

cos cot d d l si c p- si siθ d θ l cosθ c 4 cosθ 4 d d c l l 6 l l 列 O.D.E. cos si cos e 4 l e cos ; %95 d d for. %95,. 5 e %95 立數 c~6-6

4 cosθ dr r siθ dθ. %95 d d 5 I.C., 5%95 6 l 9 力 step step step d d 6 %95 d d sec %9 令立數 c~6-7

P- 4 度 9 o C 6 o C 冷 88 o C 若度 o C dt dt k T 6 T 86 o C %94 北 - d d d d d d 4 d d step step d d d step 立數 c~6-8

d, d %95 北 6 量, %94 d d d d d [ ] d 令 d d d d [ ] d d d d d d d d d d d d d d d d l l l l l c l l c c [ ] c c 立數 c~6-9

7 5 d 4 6 d 95 9 北冷 - 若 φ, c d φ, dc φ d φ d M, d N, d φ M, φ N, M φ N φ M N d d 6%95 d d. e %95 北立數 c~6 -

%95 φ, c 4 d 5%95 d 5 5%95 cos si d %95 利 d c 6 來 d d d O.D.E. d d 6 c d d 不復立數 c~6 -

7 O.D.E.. a Sow ot eact. b itegratig factor of a b. c geeral soltio.9 d d 4 A oeact secod order B as -depedet itegratig factor. C as a -depedet itegratig factor. D as a -depedet itegratig factor. 5%94 p-7 M N N M N M M N M N f f f M N f M N I e I I e I f d f d e f d f d e 9 e d d. 5%95 北立數 c~6 -

si 4 cos 4 si cos d cos d %95 北 cos d si cos si d 5%95 d d 95 d d 9 4 d d A oeact secod order B as -depedet itegratig factor. C as a -depedet itegratig factor. D as a -depedet itegratig factor. 5%94 立數 c~6 -

-4 p-56. d ± d d ±. d ± d d ±. d d d 4. d d d d dta dl d d 6%95 d d d d d d d 95 d d d d d d d d d d 6 d 9 北 d d d d d d d l 立數 c~6-4

8 d d d d d ta d d m 5. d md d m %9 北 9 d d d d d -5 P Q I e P d I I Q d c P Q Prove I e P d geeral soltio. 9 9 北 9 北立數 c~6-5

d d ta e si 95 北冷 8 ta si 95 北 9 9 北冷 9 4 d d 5%95 5 d d si 95 6 d d %95 北 cos si 9 立數 c~6-6

ta 9 d d P Q 4 d e d. 95 9-6 Berolli d d α P Q, 5 %95 α 6,, 9 立數 c~6-7

Riccati P Q R a b a 令 z %95 z l c 8 a b b a ae 令 O Neil z P Q R -5 b ae 立數 c~6-8

-7 Variatio of Parameters. omogeos P. oomogeos P Q P c e P d c P Q 令 φ P Q φ Q Q φ Q φ d c p Q φ c d p d? d 5%95 立數 c~6-9

. a b c r a, b, c 數. a a LL a a r a a, a 數 p- sperpositio priciple 9 ad are soltios of liear omogeeos eqatio a a a. Te c c te liear combiatios also te soltios of te eqatio. p- t ad t t are soltios of d t cost e t dt dt d t t e te c t c is also a soltio. %9 t t ad t d t are soltios of d t t cost e t dt dt te c t c is also a soltio. t 立數 c~6 -

- Wroskia φ, φ, LL, φ } 數令 c φ cφ LL cφ { c c LL c 零 Liear Idepedet., L.I. 若零 Liear Depedet., L.D. 7 p L p ' P o-trivial c ] T [ c, c, Lc satisfig c a c c a c c M a L c a c a L c a a a L c a te liearl idepedet. %95 c c o-trivial M M c,, L, liearl depedet cos, si, cos liearl depedet or idepedet? 9 e, e, e 9 立數 c~6 -

φ Wroskia W φ, φ, φ φ φ φ φ φ φ φ φ c 若 { φ, φ, φ } c M c M W 若 W φ, φ, LL, φ } { ad ta l ad l 4 si ad si 4 ad 94 北 9 4 d d P Q d d W. a W P W. b P d W ke Abel s formla 6,,, ad Fid Wroskia W. % 立數 c~6 -

- a b c 令 m e am bm c m m, m sperpositio priciple m m m m ce ce m m m m ce ce p m p ± iq e [ c cosq c si q] 5 %95 9 4 4 95 北 94 利 9 6 d d 5 %94 d d a b B.C. A, B 了數 b B e cos si 了數 a A e e 9 4 立數 c~6 -

" A' B, A 4B < α α ce cosβ ce siβ 9 4 " A' B, A > 4B αt t e c cos β t c si β 9 北 t 6 " 9 9 B.C. 6 9,, 7 6 9,, 7 9 si is a soltio of 4 A cos, B e cos, C cos, D si, E e si.5%95 4 立數 c~6-4

costat coefficiet, omogeeos, liear cos as a particlar soltio. Let be te order of te igest derivative a smallest possible vale of? b geeral soltio. %95 5 A liear omogeeos costat coefficiets as a soltio its order at least A 4 B 5 C 6 D 7 E 8 cos. 9 6 omogeeos liear a a a a ' a L " Let r be a root wit mltiplicit k. If k, te r e is a soltio. 95 立數 c~6-5

- b c a r step step p step p p... 4. forcig term r p a e cos φ si φ cos φ si φ a Ae Acos φ B si φ Acos φ B si φ N A A LL A A 6 4 4 e. %95 4 9 立數 c~6-6

4 " ' cos. 5%9 8 8cos 4. %9 9 d cos 5%95 d si cos cos cos 7 si %9 si cos cos cos 令 p Acos B cos C d d, d d %94 forcig term r p e cos. 5%95 令 p e cos Bsi A 立數 c~6-7

4. ta 94 北 9 9 9 北 d 5. sec d 95 北 94 9 理 9 6. 4 6 csc 95 7. e 94 e 8. 9 北 e d d t t. e cos e 9 dt dt " 8 ' e %9 c e c 令 t e, 94 9 e t t c e cte 令 p A t e t p A e d d e, 95 9 d d c e ce 令 p A B e 立數 c~6-8

cos 4 d 6 cos 4 5%95 d c cos4 c si 4 令 p A cos4 B si 4 4 Resoace %94 cos cos 6 si 95 c cos c si 令 p A cos A si B B 立數 c~6-9

立數 c~6 - r a a a variatio of parameters c c DERIVE te particlar soltio.%95 a r φ φ Cramer rle a r φ a r φ d d sec 95 北 94 9 理 9 r a a a a a r φ φ φ redctio of order 5

立數 c~6 -,,, L omogeeos p p L Sppose,,, L f f f L M LLL LLL te L particlar soltio f p p L. %95 f M M L O L L f L M LLL LLL 數 O.D.E. r c b a te differetial operator d d D d d d d d d D D D r c bd ad liear polomial operator c bd ad D L O.D.E. r D L 9

令 L m am bm c m m,m p r L D L D ivers operator 女立數 c~6 -

a a e e L a L D L a a a e f e f L D L D a a a e f e f D a a a e e L a L D L a 6 4 4 e. %95 4 9 " 8 ' e %9 a a e f e f L D L D a " 8 ' e %9 a a e f e f D a e cos. 5%95 立數 c~6 -

e 9 " 8 ' e %9 t, 94 9 e d d e, 95 9 d d " 4' 4 e 94 9 北 9 北女 L D, D cos φ si φ L. D cos φ si φ 6 4 6 cos 9 7 8 6 cos. %9 5 cos 9 4 " ' cos. 5%9 立數 c~6-4

立數 c~6-5 si cos. si cos, φ φ φ φ D L D D L 4 8cos. %9 d d cos4 6? 5%95 cos si si cos D D cos si si cos D D si cos i D e a D i D D cos si si cos si 4 cos 4 L.D. d d cos4 6? 5%95 8 4 4 5

si si si L D L D L D cos cos cos L D L D L D si 4 D 6 si. 95 si cos cos 8 D 8 9 particlar soltio cos be te igest order. %95 a omogeeos 6 b oomogeeos L D a a D a D La D N d d, %94 d d 立數 c~6-6

4 D m f e m e m f d d d t t e cos e 9 dt dt 5 secod-order ae %9 bcos secod-order a b c f omogeeos soltio ad e particlar soltio p 5%95 4 p c e c e c e 7 e 立數 c~6-7

4- a b c m 令 am m bm c m m, m m m m m c c m m m m c c l p m p ± iq [ c cosql c si ql ] d d 5 5%94 d d 4 5 5%95 Eler-Cac 8 a b %9 9 Eler-Cac Eqatios ca be trasformed ito Costat-Coefficiet Eqatios 9 9 立數 c~6-8

4 4 6 %95 a b c f r t 令 e t l D { ad D D bd D cd f } r t 6 6%95 4 P4-5 4 d dt 5%94 北立數 c~6-9

4- 令 φ Give omogeeos ad e sig te Metod of Variatio of Parameter.5%9 4 si 9 立數 c~6-4

a a a r a a a a a a m a ma a a a m a a a ta a a a cot a a e e m e m cos si 若 e a a a a e a e a e a a a 若 e a a a a e a e a e a a a m 若 e a a a m m m m a e ma e a e m a ma a 4 若 a a a a a a a a 5 若 m a a a a a a m a m a 6 若 cos a a a cos a si a cos a si a a a cos 立數 c~6-4

a ta a a 7 若 si a a a si a cos a si a cos a a a si a cot a a 5, %95 7. 95 9 9 doble root m m,m m m c e ce 5%95 8 t t %9 力 4 d d si cos si d d 令 cos φ si cos φ cos si φ e %95 北 e si φ 4cosφ e 立數 c~6-4

cos e dφ cos e φ 4 φ O.D.E. 4 φ si si d si si cos d si I e si e I si d e si d si cos e si 5 φ c si cos e φ 5 si c si d si cos si e 5 si c csc e e cos e si cos φ e d e d 5 si si c cot cos cos e e d e d c cot 5 si si si 5 si e c cot c cos c cot c e 5 si cos cot cos cot c c cos 5 si c c e 5 si c cos cot c cos cos e c cos cot c cos 5 si cos 5 si e e 立數 c~6-4

4-4 9 a a a r 若 a a a 降 4 9 4 5%94 北 4- 更 a a a a r 若 a a a a 降 5 a a a a r 9 9 O.D.E. a a a a 4-4 P Q R Q P 4 C P C 立數 c~6-44

ep P d C 令 φ C φ φ R 6 6 4 5%94 立數 c~6-45

4-5 P Q R Q t B 數 B t Pt ceck : A 數 t? d d R A B dt dt t %95 4! A c c! B c c! C c c!!! 4. 5 了度復來利來說令 t l d 數 d 立數 c~6-46

4-6 O.D.E. f, 令 p p 降 O.D.E. p f, p d d d d %95 北 dp O.D.E. f, 令 p d dp 降 O.D.E. P f, P d dp d d d dp P d 7 Fid te geeral soltio of. %96 d dp dp d 令 p d d d d dp p d dp dp O.D.E. p p p p d d dp d p l p l l c l c p c c d 降 O.D.E. c d d cd c l c l c le l c l c e c c ce 立數 c~6-47

O.D.E... Laplace. 8 5 4e t %94 Laplace 6- d f e d 9 d Leibitz s Rle 4 e t cost dt? 9 立數 c~6-48

立數 c~6-49 a d I si α α α α dα di 6-Uit Step Fctio Heavside fctio. t < > t t a t a a t t < >. < > or < > < < b a a or b b a < > < < > > a b a b ] [ f b a f a or b b a < > < < > > a b a b 7

t t dt 4 95 北 6- Delta FctioUit Implse Fctio. δ δ d a δ a a δ a d. f δ f δ f δ a f a δ a 9. δ. δ a δ a 立數 c~6-5

4. δ g d δ g d g δ d g δ a g d g a 7. δ a f d f a. δ δ 8. δ a δ a. [ f δ ] f δ f δ f δ f δ f δ f t δ t t dt f %9 t δ at b t dt 6-4 Gamma Fctio. t Γ e t dt >. Γ Γ Γ π Γ 4 Γ! N 立數 c~6-5

4 Γ Γ Γ π 95 Γ 4 Γ! 94 5 7 7 Γ 5 Γ Γ 4 Γ 4 7Γ 4 9 立數 c~6-5