Inermediae Differenial Equaions Review and Basic Ideas John A. Burns Cener for Opimal Design And Conrol Inerdisciplinary Cener forappliedmahemaics Virginia Polyechnic Insiue and Sae Universiy Blacksburg, Virginia 46-5 MATH 445 - FALL
Topics Inermediae Differenial Equaions Review of he Inroducion o Differenial Equaions Review Classificaion of Equaions General Firs Order Scalar Equaions Exisence, Uniqueness Elemenary Numerical Mehods Sysems of Equaions Equilibrium Sabiliy of Equilibrium Linear Sysems Linearizaion Dependence on Parameers: Sensiiviy Advanced Numerical Mehods
Firs Order Linear I p () p () p () r p() p () p () r d r d c ln p( ) rc p() ke r r p p ke k () p k r r p() p e e p c p() e e e ke r c r r THE SOLUTION r p( ) e p
d d e p() r p() Firs Order Linear II r d d r [ p( )] () () r r e p e r p d d [ e r ] [ p ( r )] [ e ] p ( ) r [ ( )] e p k r p() e k r e p () e r p() X () () e INTEGRATING FACTOR THE SOLUTION r FUNDAMENTAL SOLUTION e r r p p e k k () r p( ) e p
y () y() cos() y () r [ y()] y() K Classificaion s order (linear) y( ) y ( ) sin( ) y( ) nd order (linear) y( ) g y ( ) y ( ) m rd order (nonlinear) y() y() [ y()] s order (nonlinear) nd order (nonlinear) ALL EQUATIONS CAN BE REDUCED TO FIRST ORDER
? Wha is a Differenial Equaion? A differenial equaion is an equaion ha involves an unknown funcion and is derivaives. d p () p () d x () x () 5[ x ()] y( x) sin( y( x)) x p ().9 p() p() y() y () y() e y() y() y() e [ x ( )] [ x ( )] 5 5 y x e x y( x) ( ) cos( ) x() ( ) x() sin() x() WRITE WITH HIGHEST DERIVATIVE ON LEFT SIDE
? Wha is he Order? x() [ sin() x() x()]/( ) d p () p () d x () x () 5[ x ()] y () y () y () y y () () ee y y y e 5 5 5 5 y () () y () () y() e y ( x ) sin( y ( x )) x y ( x) [ x x () ( )] [ x (( )] [ x()]) y( x) e cos( x) ln( y( x)) y( x) ln(cos( x)) y( x) ln( y( x)) ln(cos( x))
4 h? Wha is he Order? x() [ sin() x() x()]/( ) s s d p () p () d x () x () 5[ x ()] nd rd y () y () y () e 5 y() y () y() e 5 s nd y ( x) sin( y( x)) x x() ( [ x()]) rd y( x) ln( y( x)) ln(cos( x)) n h order ordinary differenial equaion n ( n) d y() ( n) y () f(, y(), y(), y(),..., y ()) n d
s Order Equaions s order ordinary differenial equaion x () f(, x ()) f(, x): R R R d p () p () d x () x () 5[ x ()] y() sin( y()) y y () 5[ ()] cos() f (, p) p f (, x) x5x f (, y) sin( y) f y y (, ) 5 cos()
s Order Auonomous DEs x () f (, x()) x()) d p () p () d x () x () 5[ x ()] f (, p) p f (, x) x5x Auo Non-auo y() sin( y()) f (, y) sin( y) Non-auo y y y () 5[ ()] cos( ()) f y y y (, ) 5 cos( ) Auo () [ (/ ) ()] () p r k p p Logisic Equaion Auonomous f ( p) r[ (/ k) p] p
? Wha is a Soluion? p() p() CHECK p() e p() e p () e p( ) () 7 p e p () [7 7 e ] p( ) x () x () x () cos() p() ce CHECK x () sin() x () sin() x () x () 4cos() x () 4sin() x () 4cos() x ()
? Wha is a Soluion? y() [ y()] CHECK y () y( ) y () ( ) y () ( ) y( ) ( ) ( ) ( ) y () 4 y () 4(4) 4 y( ) 4( 4 ) ( 4) 6( 4 ) 4 4 y() 4 y ()
(Σ) Definiion of Soluion d x () f, x () d A soluion o he ordinary differenial equaion (Σ) is a differeniable funcion x():( a, b) R defined on a conneced inerval (a,b) such ha x() saisfies (Σ) for all (a,b). d x () () d x () x TWO SOLUTIONS xl (), xr (),
Soluions 5 xl (), 5-5 - -5 xr (), - - - - 4
Iniial Condiion xl ( ) 5 xl (), 5 xr ( ) -5 - -5 xr (), - - - - 4
x() f(, x()) Iniial Value Problem s order ordinary differenial equaion x( ) and an iniial condiion x x() x() x() 7 and x 7 x() ce 7 () x ce c c 7 x () 7e ALTHOUGH THE DIFFERENTIAL EQUATION HAS LOTS OF SOLUTIONS, THERE IS ONLY ONE THAT ALSO x() 7 SATISFIES THE INITIAL CONDITION. x() x()
Logisics Equaion (LE) p () r p() p() K Kp p p K p p() p () r e R 9 K 8 7 6 5 4 5 5
Iniial p : < p <, 4 r.96, K 9, 5 x 4.8 p, p 5,.6.4..8.6.4. p p 5 75 p p p 5 5 5 K
Firs Order Scalar Equaions x() ax() cos() x () r[ x ()] x () x x x x (){[ ()] } cos( () ) 5 () x () OR cos( x() ) 5 x() [ x ( )] FIRST ORDER SCALAR EQUATION x() f(, x()) f (, x): D RR R RIGHT-HAND SIDE K
Firs Order Scalar Equaions x () ax() cos() f (, x) ax cos() x () r[ x ()] x () K k f (, x) r[ xx ] x () cos( x() ) 5 x() [ x ( )] f(, x) cos( x ) 5x [ x] [ ] x( ) x ( ) x( ) cos( ) x() cos() x () [ ] x( ) f(, x) x cos( ) [ ] x D (, x):, x
EXISTENCE AND UNIQUENESS
(Σ) Soluions o Firs Order Equaions x () f x, () f (, x): D RRR A soluion o he ordinary differenial equaion (Σ) is a differeniable funcion x():( a, b) R defined on a conneced inerval (a, b) such ha. (, x()) D for all (a, b),. x() saisfies (Σ) for all (a, b). x () x() x () () x x() HAS TWO SOLUTIONS e k ( ab, ) (, ) k x l ( ), x r ( ),
Muliple Soluions x l 5 ( ), 5 ( a, b ) (, ) r r -5 - ( a, b) (,) l l x r ( ), -5 - - - - 4
(IVP) Iniial Value Problem (Σ) (IC) x () f x, () x( ) x R f(, x): D RRR A soluion o he iniial value problem (IVP) is a differeniable funcion x():( a, b) R defined on a conneced inerval (a, b) such ha. (a, b),. x( ) = x,. (, x()) D for all (a, b), 4. x() saisfies (Σ) for all (a, b).
Iniial Value Problem (IVP) (Σ) (IC) x () f x, () x( ) x R f(, x): D RRR R x x() D
Iniial Condiion x () () x() x x l 5 ( ), 5-5 - x r ( ), -5 - - - - 4
Exisence Theorem (IVP){ (Σ) (IC) x () f x, () x ( ) x R R x y() x() D f (, x): D RRR Theorem. If f: D ---> R is a coninuous funcion on a domain D and (, x )D, hen here exiss a leas one soluion o he iniial value problem (IVP). / x / x () () f (, x) x D R R f(, x) is coninuous EVERYWHERE!!!
Non-Uniqueness x () () x() x / x () x () {, ([ ]/ ), c c c R c c!! INFINITE NUMBER OF SOLUTIONS!!
Uniqueness Theorem Theorem. If here is an open recangle D abou (, x ) such ha f (, x ) and (, ) f x x are coninuous a all poins (, x), hen here a unique soluion o he iniial value problem (IVP). R x x() D