1 st order ODE Initial Condition

Transcription

1 Mah-33 Chapers 1-1 s Order ODE Sepember 1, s order ODE Iniial Condiion f, sandard form LINEAR NON-LINEAR,, p g differenial form M x dx N x d differenial form is equivalen o a pair of differenial equaions dx M x, N x, d d M x, N x, dx Separable Tes for Exac Inegraing Facor pd e M x dx N d M x x, N x, Genearl soluion: General soluion: There exiss f x, such ha c 1 g d M x dx N d c,,, df x M x dx N x d f M x,, x f N x, Soluion of IVP : Soluion of IVP x General soluion f x, c 1 s g s ds Equaion wih consan coefficiens: a b x x M s ds N s ds change of variable reduces homogeneous equaion o separable Soluion of IVP x : x,, x M s ds N x s ds muliplicaion b inegraing facor makes equaion exac General soluion: Homogeneous equaion Inegraing Facor ce Soluion of IVP : a b a a b b e a a use change of variable ux x v d udx xdu or dx vd dv back subsiuion u x v x M N x if h x N is a funcion of x onl hen hxdx e ba, graph in a case of a, b Equaion is homogeneous if, n, M x M x, n, N x N x To check ha (x) is a soluion, subsiue i ino To check ha x is a soluion, subsiue i ino if N M x M is a funcion of onl g hen d M x, N x, dx dx M x, N x, d g d e

2 Mah-33 Chapers 1-1 s Order ODE Sepember 1, Classificaion Differenial equaion ODE PDE Order of ODE Normal ODE Linear ODE Non-Linear ODE IVP Soluion Exisence of a soluion Uniqueness of a soluion

3 Mah-33 Chapers 1-1 s Order ODE Sepember 1, s order ODE Normal form f, d Normal equaion is an equaion explicil solved for d Slope of he angen line o he soluion curves a he poin, f,, equaion of he angen line:, f, Direcion field (1.1 #1) > resar; > wih(deools): d 3 d > DE:=diff((x),x)=3-*(x); DE := d d x ( x) 3 ( x) Ploing he direcion field wih he soluion curves saisfing he iniial condiions: >DEplo(DE,(x),x=-..,=1..3,{[()=],[()=],[()=1.5]},color=black); Mehod of isoclines

4 Mah-33 Chapers 1-1 s Order ODE Sepember 1, Linear ODE Inegraing Facor Sandard form p g Iniial Condiion: Inegraing facor e pd General soluion c 1 g d c 1 g d (for numerical inegraion) Soluion of IVP: 1 g d Case of a consan coefficien: a g Inegraing facor General soluion a e a a a ce e e g d Soluion of IVP: a a a e e e g d If g b a b General soluion a ce b a Soluion of IVP: a b b e a a Exercise: Solve 4 1 and skech he soluion curve

5 Mah-33 Chapers 1-1 s Order ODE Sepember 1, 17 5 Example: Find a general soluion of equaion co x sin x and skech he soluion curves. Soluion: The inegraing facor for his equaion is x e coxdx e ln sin x sin x Then a general soluion is c sin x 1 sin x sinx sinx dx c sinx sinxcosxdx sin x sin x (double angle formula) c sin xd sinx sin x sin x (u-subsiuion) c sin x sinx 3 Maple: creae a sequence of paricular soluions b varing he consan c, and hen plo he graph of soluion curves: > (x):=*sin(x)^/3+c/sin(x); ( x) := sin( x) c 3 sin( x) > p:={seq(subs(c=i/4,(x)),i=-..)}: > plo(p,x=-*pi..*pi,=-5..5); x x x

6 Mah-33 Chapers 1-1 s Order ODE Sepember 1, Separable equaion Differenial form of ODE M x, dx N x, d Noe ha equaion in differenial form has no disincion beween independen and dependen variable differenial form is equivalen o a pair of differenial equaions dx M x, N x, d d M x, N x, dx Separable ODE M xdx N d General Soluion M x dx N d c Soluion of IVP x x x M x dx N d m Equaion is homogeneous M x, M x, m of order m if N x, N x, are homogeneous funcions of order m Homogeneous equaion can be reduced o separable Back subsiuion: b a change of variable o ux d udx xdu u x or b a change of variable x o x vu dx vd dv x v Implici soluion Inegral curve Suppressed soluions Example Problem. #1, p.47: Solve Example Problem. #31 subjec o iniial condiion x 3

7 Mah-33 Chapers 1-1 s Order ODE Sepember 1, 17 7 Example Solve he differenial equaion x x x dx xd differenial form Soluion: M and N are homogeneous funcions of degree. Change of variable: ux d xdu udx u x x dx xux( xdu udx ) 3 u x x u x dx ux du x 3 u 1dx ux du separable dx udu separae variables, x x u 1 d u 1 dx 1 x u 1 inegrae ln x x ln c ln u soluion u 1 c back subsiuion u x x x c general soluion(implici) Check for suppressed soluions: earlier we assumed ha x, hen check ha x is also a soluion bu his soluion is a paricular case of general soluion when c. Use Maple o plo he soluion curves: > f:={seq(x^*(^+x^)=i/8,i=..1)}: > impliciplo(f,x=-..,=-5..5) or plo level curves of, f x x x

8 Mah-33 Chapers 1-1 s Order ODE Sepember 1, Exac Equaions and Inegraing Facors Differenial of f x, is d f x, f x, f x, dx d x Exac equaion M x, dx N x, d is exac if here exiss d f x, M x, dx N x, d some f, x such ha Tes on exac equaion M N x General soluion f x, c level curves of he surface defined b f x, f x Finding f x, 1) M x, f ) N x, f x, M x, dx k d d M x, dx k N x, find k 3) General Soluion: f x, M x, dx k c Inegraing facor μ Equaion muliplied b an inegraion facor μ becomes exac. M N x i) if hx hen x e N h x dx N M x M ii) if g hen e g d

9 Mah-33 Chapers 1-1 s Order ODE Sepember 1, 17 9 Example (p.95) Solve x x Rewrie in differenial form Tes for exac: M x dx x d M N x x x x N Exac Find f x, 1) f M x f x x f x x k ) f N f dk x x k x d dk x x d dk d k c Therefore, f x x c x x c c 1 combine consans, hen General soluion: x x c

10 Mah-33 Chapers 1-1 s Order ODE Sepember 1, Differences beween linear and non-linear equaions Theorem.4.1 (exisence and uniqueness of he soluion of IVP for linear 1 s order ODE p g ) Le, and le p, g C, (coninuous funcions) hen he linear differenial equaion p g has a unique soluion such ha,, Theorem.4. (exisence and uniqueness of he soluion of IVP for non-linear 1 s order ODE f, ) Le f, f,, C,, and le, and, (coninuous) hen he non-linear differenial equaion f, has a unique soluion, h, h,, h h h such ha. Here, h> is some posiive number. Remark: if onl f, C,, is coninuous, hen soluion of IVP exiss bu is no necessaril unique. Noe: Consan soluions f x, if he heorems guaranee onl ha under given condiions here exiss a unique soluion of he IVP, bu he do no claim ha he soluion does no exis if he condiions of he heorems are violaed. x g x, if f x,b, hen b is a soluion g a,, hen x a is a soluion General Soluion F x,,c Soluion which include an arbirar consan. Suppressed soluions General soluion of linear ODE includes all possible soluions (complee soluion). For non-linear ODE, here can be some addiional soluions. Soluions no described b he general soluion

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12 Mah-33 Chapers 1-1 s Order ODE Sepember 1, Auonomous Equaions f independen variable is no in equaion explicil Criical poins k : k criical poins k are consan soluions The are called he equilibrium soluions. Second derivaive chain rule df d df df f f d d d d Hpercriical poins k : k inflecion can occur onl a hpercriical poins second derivaive es firs derivaive es concave up soluion is decreasing phase line equilibrium soluion (sauraion level) concave down concave up soluion is increasing inflecion poins concave down soluion is decreasing equilibrium soluion Logisic Equaion 1 r 1 K (7) criical poins: Firs derivaive f 1 r 1 K and K Second derivaive df f d df r 1 d K hpercriical poins: 1 r 1 1 K K, K and K

13 Mah-33 Chapers 1-1 s Order ODE Sepember 1, Sabili of Auonomous Equaion sabili of equilibrium soluions k, k : k f semisable unsable asmpoicall sable unsable Example:.5 # 1 d d

14 Mah-33 Chapers 1-1 s Order ODE Sepember 1, Mahemaical modeling Tank problem (1.1- #1) m m1 min m ou mass balance r in Qin Qou Q Q Q Q in ou mass balance for process during ime inerval Volume V Q dq Q Q in d ou Q g g Q r c min mass mass flow rae r ou dq c in r in c r d ou ou g c gal c ou Q V gal r min concenraion volume flow rae V gal volume for r r r cons in ou and V cons dq r Q c r in d V Exponenial deca (1.-1,13) dq rq r d Exponenial growh dq rq r d Populaion model (1.1, p.5) dp rp k d r, k du Newon s Law of Cooling k u T (1.1-3, 1.-15) d k

15 Mah-33 Chapers 1-1 s Order ODE Sepember 1, Exponenial deca 1 r deca rae, r > sec Exponenial growh 1 r growh rae, r > sec dp rp d dp rp d Tank problem Q lbm amoun of sal dq r Q c r in d V Populaion model (mice-owl) p mice populaion 1 r ear growh rae reproducion rae dp rp k d mice k ear coninuous rae of killing mice p Bank model p mice iniial populaion S $ invesmen or deb 1 r ear annual ineres rae rae of reurn $ coninuous annual k ear rae of deposis k 1 m ds rs k d k S S e e 1 r r r 1m S S e e 1 r r r $ coninuous annual w ear rae of wihdrawls w 1 m ds rs w d w S S e e 1 r r r 1m S S e e 1 r r r $ coninuous annual p ear rae of pamens p 1 m ds rs p d r p r r r S Se e 1 S S e e 1 r 1m r $ monhl deposis, m monh wihdrawls or pamens $ S S iniial deposi or a loan 1m S S e r r 1m r Newon s Law o u F emperaure du k u T d 1 1 k sec cv p sec > ime consan ha Falling ball v f veloci s f v v iniial veloci s m Ff Fg v mg dv v g d m

16 Mah-33 Chapers 1-1 s Order ODE Sepember 1, Falling ball For convenience, assume ha he posiive direcion is UPWARD (in conras o exbook, see p.) v v v v up down Ff v direcion of fricion force is opposie o direcion of he veloci m v 1 v Fg mg graviaional force is direced downward (negaive sign) erminal veloci v mg v erminal veloci F F F g f resuling force dv F m d Newon's Law h max h when v max h = high dv m mg v d governing equaion for veloci v h dh v d inegrae h h v d 1 dv.3 # m v d v v mg no fricion force dv.3 #1 m v mg d v v fricion force is opposie o veloci, auomaicall changes direcion.3 # Par I dv v m v mg d 1 h h v d 1 v v fricion force is downward dv Par II v m v mg d h h v d max 1 1 v 1 fricion force is upward

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18 Mah-33 Chapers 1-1 s Order ODE Sepember 1, Inegrals Expeced Known 1. sec x dx ln sec x an x c. csc x dx ln csc x co x c 3. an x dx ln cos x c 4. co x dx ln sin x c 5. ln x dx xln x x c 1 x 6. a dx 1 x arcan a a c 7. 1 dx arcsin x c a x a ax 8. e sin bx dx ax e asin bx bcosbx a b c ax 9. e cosbx dx ax e asin bx bcosbx a b c 1. sin mx dx mx sin mx cos mx m c 11. cos mx dx mx sin mx cos mx c m