Chapitre 2 : La résolution des équations différentielles
Transcription
Chapitre 2 : La résolution des équations différentielles
! ! " # $ % ( &' ) * * ! " " % ( # + + + &, &. - . . + * . ,/01, 23 5 $/6 # /-'7 0 4 8 9 :9 +& / +1α4& & $5 , ! 8 "! # / " % ; ( # # 5 < & -- . #$ ** * . : * . . * * = # 5 % . . $5* >* $ + . / + ' () *+ ? * ! #& " ; ; ! ! ! ' , - %. # -$ -- * @ @ A @ - " , @ @ $ / # & @ - @ * * * B , * # * # C ** 5 * ; -- #5 # * D ** * < @ @ 4 -- # - < -* 1$/ 4 ** @ * -- ** 5 * E < # * 5 ( 1 @ A = * $- @ - () 18 3 1 ** B * - -- * = ) #* $/6 / * -$/6 * < $/6 # $ "4@ $/ ** # # 7 ) 4 1 ∀ ∈ ⊂ @ 4 ( )= ( ( )) * # 7 1 4 1 4- # * * * F 4 > ( )= 6 & * $ < - 1$/64 < - $/ # 6 - D 1 4 1# * & - * - - - # - -- & * < 7 < * * #* @ < * -- # * - # * * ** < * 5* *@ B * * * * 5 7 < # @ 4 & & ( ( ( ) ( )) = -< 1 > # * - * 6 * - # * * -5 5 * * = @ * # *@ * # = * - $/6 # = * < ( ) $/6 # = 6 # $/6 ( )= @ * $/6 × ∀( * 1 ∈ # 5 1 4 1 4@ × -- 4 * / D 1 - G = ( )= & 4 7 ) ( )∈ ( × ** )− ( & * 1 H.%;I 4 > > 5 − 1 4 A & < J - )≤ # -- # * # 5 & * / $/6@ 6 * + ( ) ** 5 − = * * * # # ( ) -5 ( ) = * * -5 @ # * 4 - - G 6 < * # & ( ) 1 * * * 8 ( 6 > # 6 ( ) * * * - * ) + 7 * < * - + Φ # 7 + Φ( = ) + - 1 !4 - - # @ # * [ # + ] $ * α * +α + α ≠ - +α + 7 (β = + − >@ + +β + + − ) 1 "4 < * + 2 * $ 5 - F -- / * @ -- - & * * 8 5* * & * * * - # * # * ** @ # # -- 5 * 5 # * @ * * & < * < # #- * * @ # # 7 # 5 - @ F * % ! 0 % * * $ #5* # * # - 7 ( )− → = * -- > F * * * * 5 @< * * - G 7 < ! ** 5 * * : - 1 %4 ** # ) < < * * # @ F < * @ # * ε * 7 + Φ( + Φ( = + = + 6 ) 1 ;4 )+ε @ # 5 * @ 7 − ≤ − * * 7 ε + 1 * * * * # B # * # # % * * # * F * # *- % $- @ * - * # # # (4 -- * 7 < * # ** #** 5 < @ # - - * # *G * < * 7 ( = + 6 ( )) 1 4 1 4 7 = → + Φ( )− ( )− @ A* @ D - - * * >- * D * # 7 - G # @ * * * - 7 1 # 5 4 A # # 6 7 ≤ KE < : * * # @ *@ 1 * - − = # 6 < # * # 4 & # & " @* # - # * ** B * & * @ * 7 - # * * # * * H.%;I @H LM((I H LM;!I* @ # & * " * $ 6 - ** 5 )≈ + − + $ * + )≈ -- + * * # 6 # @ # * & + ) 1 5* 4 @ N #- -- 6 ? & 0 =− * * & * 5 # * * 4 > * * 6 + 5 # 5 =− * * * # # 5* * ( + < * * # = 5 @ * #** 5 * * 1 47 − + F 5- 5 ) * * 1 )= ( ( + * ** 5 - = + < ** 5 + & 7 ( )= ( ( - # $ = -5 -- * 7$ 5* O- * * < *P # * & F 8 @ & # * * = * = # $ ** 5 % # # # 0 − = 5* * * -- * * 6 − & 10 10 / = - !% " @ * * * * 7 * * * # A * # A % * - # * < − * + @ =− !% 0 6 * * =−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∈ = = 1 7 + + & < - = 1 ;((4 * - - L # 1 ;( 4 # ;4 * # # / * @ $ @ @ &' @ /3 − − + $ @ @ L + RL 3 @ /3) $ # @, -- • # ≥ @ - + • # = @ - + H , ! I • # ≥ @ - + H , % I * * @ * 7 '$ # # * # * ' 1 * ** * " # + 4* C & <@ # < * 5 N* & H L "%I 7 # 7 < # $ -- @ # * * ( &' S? # H , ! I ? * 5 - *@ @ # & @ '> # 1 * < # ? ** * # 0 # * # * * = <( * ? Q @ # # λ < # - * * * * * * 4 . ** D E C * #5 * 1* = * # $ ( − @ λ= ' * * $ * − * ( * '4 N ? # *5 @ 5* 10 - $ -- #= # * # 5% * & * 7% /3 : / @ # * * < #5 Q - @ D - * * # * # * < * # <* $ ** # - # * & *& < 5 $ # * 5( 6 5( # ' # 5@ 5T * * * * 5 *@ # <7 = J # * + < 1 − 1 4@ * − 6 * + = + + 4 5 7 1 4 1 4 7 * + 4 * + * 1 + + # # $- @ # + = * = $ − = < + + + 7 + + @ * − * # * * < * - G # 4 1 !4 * # * 1 ' * 1 + < # − − + # ** + + ** 5 = * − + − = #- * 4 . ' * - 1 ; "4 *& # * ** @ @ * 5 -- - N * < # C B * # # ** & & & B / = #** -- * N * * F * # # * # 1 4 − − − − − − − − − . 1 4 / & 1 4 ; # * # # * 1 * 5 4 # - $ 7 - * -5 * $ -- @ -- & C @ > -= $ < * / 7 - @ * * * - @ 7 1 # &- * *F * * # * '@ * E < & * @ & * * * * & * < * * ' * * * 5 ( & * * * E < -- '* # * * -- & # - $ @ * & * < * 5* ** * # 5* / 3 @ - 7- # # # * , 6 * & & H C ;%I #5 - # * < * * @ & <* * $ # = - * #** * * * -- & * * 7 * 2 *& @ 'H C ;%I D # * - * β α -- 1 "4 7 α 5& + +α + +α * * ! %4 - + - $ + * * * + ≈ * # "4 * @ 7 = / 1 < @ * * − = = + − = (β + + +β + − ) @ = / -- α 1 * 4@ β -- 1 5 * # + 4 8 ,/0 " * U + < < - # # + # * ) - * * @ ** 5 5* = * * * + * * & 1 ;4 + # 6 # + - 5 = * # < + - & B 5 2( * ) * # * − ! < * 1 !4 > * @ ) # − + 10 7 ∇ = ) $ γ ∇ γ = − 1 (4 - G & = − + * ) 7 − ! − + − + ∇ * * & * ;4@ = - − + 10 / + -- * 6 * ) ** 5 * * * - 7 + $ 0, ' ) #* ∇ <* 5 =∇ − -- −∇ − - ( = - − 7 " ( = # # ) ( − 1H LM((I 4@ E < # * # * − # + γ + − + F * * * F > * # # 8 - C * -- & F * * * ? # = + = + = + = + = + − − + = + = + − − + 5*& * -5 N * - + − * < = % * # $ & * * @ -- = @ * - '$ 5* * ** * * 5 , 7 8 ** N # < # + 4 # * # = 1 * * # # # / * - * − " * − − ( − − 9 2> # * * * # * # + * ) & * * # + * - * + @ @ - 7 @ 10 "4 # < * , # * % * γ = + - * @ -- γ -- 7 γ + γ * ) − + 5 '> 7 ! − + − 10 + / = <* γ ∇ + = -- γ -- # = = − + # + , + γ − # + 6 + − + + 4 1 4 @ γ = + . C = @ * = @ # $ * * *A = + = + = + = + = + = + + + − = + = + + + − " * > # + + + % − + − − 9 @ # + * - 1 1H LM((I 47 γ + γ * > γ = − + * # + & ) F 5 + − − + . * -5 S ># * 7 < * V 7 7 - * - # * # + , - F ** 5 E < ** 5 * = + + + + & 7 + 7 V * # + + * = + + + + ; 6 $$ 1 6 * $4 N + #** @ @ # * # *- & & + * - * < * $1 -5 1 $1 $484 * W -- 4 & # > (41<( %?@ * - # * * E * % = -- # + 5 ** 5 * "′ 0 1 * * " ) 4 = + 6 − + / * @ >@ # * 7 + 1 4 1 4 1 !4 " - G " + − − + + − + − 10 @ δ ∇ = + = ( − + δ = − δ 7 * * 7 = ,/0 # + - + δ = # " .: * -- + # * ≤ δ = * 7 # < ,/0 C * * * > ! * @ ≥ # * * ! * ,/0 * & = + = − = + − + + = + − + = + − + = + − + = + − + ' = − + − − − = − − − + − − − − − − − + − − = − − + + − − = + (419 ** @ - 1 * # # W @ 4 8 # * 1 1 * ( ) - #** "4 & # * & #5* 8 * 1 #5* * & * * * * 7 44 > -@ * # * - = 5* + + 45 " $/6 # * = − % > # + # * # ' * * : * : /3 ** '1 = +. A 4<1 4@ 7 + = = + + + 1 = + # = + = #** 1 %4@ %4 = + = + = '< $/6 7 $ & = + + + + = + = + + $U 8 9 - * 1 * @ ;4 - (4 1 4 * * 7 = 6 1 5 -- "4@ ;4 X * 1 1 = * * + % = $% -#** = + = $= $ = * # @* 5 * # - 1 ;4 = + + + = + 8 = + + 9 * $/6 # 1 W > 4 + H LM((I 5 α + − α + − = * 1 (4@ * , & # A = F β = = β = - G '7 + − 1 = - - = * * < + # 6 * 6 * = + = ( + − < 1 + − + − ;4 ) = 5 @ * - *- α + − = α = = # 7 β = αα− 4 = ( + − + − ) 1 β = = β β− 4 * @ # * $ * * 1 :9 & @ − + ) < -- = # + − + − − − + − C Y 1 4 + − + < " * − = − - :9 &Y - − − − ** * * = + = + = ( + + + *7 ) + ( + + + 1 4 -- & ) - 8 - # O P7 * # * / * ** 6 * * * # * @ * 7 ? # / * < * # * # @ + H L ;;I* + * + A * , + + − 1H L ;;I 4* - * * & > - 4 ** 5 :9 = − * # < * * :9 + 1 # ** − ( = − 7 * 5* + ** $ * + 1H LM((I 4 $ (4 :9 + > * ** * − - 5 N * * @ - * * - 5* F - * 1 < * *@ *5 * # * * & * 4 * * * < ( &! = * @ # ** * <( @ < - * * @ * 5 @ # * * # * 0 =− =− * ,/0 * F ; 7 Z * # 1( ** # * 2[ > ** B <* * 5 * $ * 5* * @ '@ $ 5* $ ( " ( %* & '4 * @ @ * # * A * * / @ 5* F * * * * & * 10 /3 B 4 0 % (419 " 7 9 % 7 =− - * # > N * - # * @ * 1H.%;I 4 < @ 1 5* * =− !% 5* * * 4 % 7 * - * 5* < * < & 7 - @ 6 * * & # * * * * - # $/6- * 1 < * -- 4 * @ * X * & * * - 7 +& *2 +# 5 ** +& & * * * @ - U : #** 5 ϕ ** - ** 5 -- ∂ ∂ < @ = −ϕ @ 1& ** $ ** - H LM;!I7 ( ϕ ( )) ( ) = & ( ) ( ) * #** 5 4 = - ( )= 1 # # $ 1 * U 4 / * @ U @ 7 =( + &) 1 * + λ - - * * < * !4 < λ "4 &@ * # & 7 ( )= =$ $∈ 1 =$ * A %4 & ** 4 * * < * @ * -- & - & < + ** / =' 1 ;4 =$ * ' 5* * * ) ) * * - * < * ! (ξ ) = (α − β ) ξ + * *5 7 ) + (α − β ) 1 * *5 - 1* 4 * @ * < # * # # ? # C 4 < % % =9 < 0 = 0 '> @ * 90 % 7/3 = * 0 ;7* " < 7 $ 5* =− N F @"% * '$ # * ** J < # * * & @ * @ * 5 # $- @ B # 2 % 7 * * ! 4 # $ =− @ # * < '$@ 1 6 #5 < # % 7 * !ξ - '$ # '>$ * 10 # ( * * * < ) (4 * 1H LM;!I 4 0 * @ # 1H LM;!I 4 * @ # < ' *5 * * # $ * * * & *5 D * * * 1 # * %4 # # @ * &* 6 +&: '$ 5 '> +& * 8 5 # + &, 0 - 5 * = * # + &. # $ = @ N* = * 6 = N* * * # *5 < 0 $ 5* @ *A F #* G # * +& # + &. * = 10 10 4 4 " * " * $ * " * @ * 2( 2> 2( ?D 77 ?D 77 E ! ? ,/0* *5 1 @ 0 4 * = @ * +& A * N* #5 & *<α α=2 : * # 0 # * +& @ $ * ,/0 # * @ * # " 10 4 * (417?D 7 F7 ' +& 9 N* @ - 7 (α - +1α4& / * 7 { ( −µ ) < α (α = µ α 0 % * 1 +1;(\ 4& µ≠ - } <α <π @ & ,/0 α ° 1 4N +& +1α4& ° ° ° 4 7 ° ° 2 >* * B 5 * &* ] *5 < * 5 '@ . # * : 5 * J - 1 7 '( )) = ∀) 6 - →−∞ '( )= * @ '( →∞ )= = ) )→∞ * @ * 1 A1H LM;!I 4 * * % ;4 * #5 '( ) H LM;!I 7 1 1 "4 O- * * 7 * 4 #5 P 4 ( =− * * ( )) ( ) = − * < 0 # $ * *@ * < #- @ - * - & - & '( = * = < # = # →∞ $ * * - @ F @ 10 + 4 = = * *A ' 1 +& 7 )= 1 * - * 4 * & % 0 +& * - - G 6 1 ** = # $/6 * N A ( / * @ * ∀ < * +& ** & * - & 47 )− ( ) @ − ≤ν − # $/6@ 1 - !4 7 ; − ν≤ * 1H LM;!I 4 ν≤ !4 @ * * ≥ * ,& ≤ @ - − 1 * # * 5 * %4 * * @ ! & 7 * %4 "4 - &' * 1 5 # * ∀ ≥ 7 − 1 ν − − * * # $/6 6 1 ≤ F * ** * * - * D < - ,& * * '6 * H LM;!I * * 5 *@ $ - * # * ,& ,& +& ,/0 # O< * P ( > # * -- - C - @ +1α4& - $ -- @ * +& * * * * F < * - 5* &' &, - + 2^ < & & & 6 6 8 6 8 8 2_ # * - +1α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∀ = 1 . % = !% D G 7%= /3 !% D % G ;4 !% D 7 = /3 !% D G 5 #5 * - ' * < S * * - 5 # * * * -& @ . - 0 2 > 5 5 # *@ @ #* -* F 7 * ** $/6 @ #* & -- * * & * @ < * " * 1 4 - -5 - G −* − λ = λ * **& 6 - & " = " − −λ " 1 !(4 ** 1 ;4@ $ < * 6 7 * * * @ #5* @ #5 = / #5 @ +)= N +)− ) * * & @ 7 ( = ) )− λ ( +) − 1 ! 4 7 =− λ )= − − )λ = +)− - # $/6 ` * * 1 - D 1 ! 4 # -- * 5 5 7 )= # 5 * -- 7 - G - -- = @ a@ @ $/6 * * -- =ϕ : #* - = λ 4 * H = * #5 * 1 ! 4@ # - $/6 = 6 #- ** # ( # & )= * # " # < 47 ( * 1 ! 4 1 # * * * # ((I * # * > > . % & J L - N * # =∂ @ ** * L - * * # * @ + # * ∂" & # 1 ;4 6 ** $ " = @ , L @ "− "" + @ - # & 1 (4 <7 =− ∂, ( ∂" ") " = − ∂, ( ∂ ") * 1 ! 4 * O PN - @ L * F ,= * * # - # 1 ! 4 N . L * '= ′( = * D ") '&' = & - &= , - * 1 #5 1 !!4 * -- A − + + − − B # &' * @ C ; − 7 * − + + − − + + . & # * − 7 > $ ;4 & * * * * -- − ** " * 7 < # & * * ** * @ # 7 F - 1H LM((I 4 * +* * 7 # * = % H IJCCK -- # - 6 0 # 1H + * '$ # * F * $ - ;"I 4@ # &<& * @ * < # * * 5 * # & * * * # < * 6@ *4 & * * - = * * & F * * - - -- & * - # 1- & 6 * F J * * & < > - . # @ * * O # P7 @ 1 * − @ # * * # * - ** * 4 - * * & * * @ -- ↔− # 1 !"4 #* O = P - % # ↔ 5 < Φ = = #-5 - *@ & = # # $ 5*& @ *A * * > * G% * * 7 & D & * # (Φ ) * Φ = Φ −− Φ− = Φ = Φ −− @ * * % & F * * * < 7 - $ - & * Φ 6 < # Φ ! D & * < - # * * < * # * = D < > * * - =Φ 7 * = 1 * 4 * -- N & - * * = H A ; I 4 * # * * * #* - 1 * H L ;;I @ # # $/6 >- -- * * F * * #5 5 5 * * H L ;;I 5 @ * * = & * = 5 @ * * - @ # 7 7 5 * * 7 * < @ K * * ** * @ * = * & * 5 >- >- E * * = * # * = * > * * * * - & * / % " # 5& @ - 5 & -- < & # $/6 # # 5 @* 1 # / 5 5 * - * * * @ * - & * * & - * < +- * - & 4 6 7 "=# " #= " # − - " λ = " "= " # 5 " @ λ= λ " @ #= " : - @ λ 1 !%4 -=∂ * ∂" # 5 7 " -" / - G @ * # $/6 # <@ # # # F # # λ = λ 5* "# −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b @ & * - * # D = - - < # * # N 5 * F & - # / # # * * # & * # & " * # # * * # @ * @ $ # = - -- * * & ** * # # * -- * # * # * + U 2 # @ * * * - $ * @ * < * 2I # * 1 4 - * < * * * - & 4 / F @ -- * * # H /K;!I @ - * *F # * 5 * # 1 #- /# * < H L * * D * = * & - #* # @ # * # N E < * * & * > * # & H U ; I* * & 7 * < * * @ # < & * < @ < * 1H C ,0%"I 4 & - # E < 5 * * -- $/6 *1 * - * * $ -- @ # - ** * P J * O - * & # * & < # -P - & * - <- - $ % 5* * # $/6 #* 4 > * O 5* * 7 +∆ = (∆ ) 1 "(4 J * # * 4 $ - < 1 4 < # ** -- < * * - +∆ < # J F * 5* - * * # > * -- < - * * -- # $ # * * * = < -- * -- -& - ?F 1 "(4@ # $ # * $ @ ?F * - ** @ * # F 5* # # * & * * < # * # 6 * < #* * * 5* * ** 5 -- @ + - +∆ * < # * * < * $ - # < #5* * # * # $/6 #5* $/6 ; - * > * * #- * -- 1 * > ) * 4 % * 5* - * 1W 1 O 4@ - P 8 * * 8 # * * < # @ # G * # $ * # # -- M 2 * H ,M;%I # < # Z # * @* - , & & 7 = ; 7 # − − ∂ + ∂ @ 5 5* ⇔ = 1 " 4 * 51 * & H LM;!I 4 ; J$ * #* # - 0 #5* * $+ $ * < $/6 & 5* * * # $ - 5* * @ # * * # > # * < < * = + + ** # 7 + @ 1 & + 1 " 4 + -- + ** 5 C - 4 / < - G @ 7 − − ∆ * ∂ − ∂ ∂ ∂ − -# * - < - # 2 # * 7 5 7 #- # * 2 @ D@ a4 * @ & > < 7 # @ > # -- 5 & * * * 1K @ # - G * 2 & & * - *- < - * 7 > + * = * & * 8 3 4 6 - + = * 1 " 4 - 5* < * ∂ ∂ + * - 1 − ∆ = 1 * ** - -- 4 < . & * & 5 5* * #5* < 5 * - * !( * 7 K = @* & > - 1 " 4 * * 7 ** # * @ # *- -- * / * @ * # # *& =& - * < ** 5 * @ -- F * -- * * * = - * ? * * * # * * 5& & @ F * # * * # * * * < # • # * 7 * • - •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c$5 < P * * * $ -- @ >- # * && 1$ & * * * = c < * -- * * * 5 && c 4 #c$ & * & * * c 4 # * 5* < 8 * X * * * - * # #5 57 # * * / * @ # - 5* < 5* * * * * < * X & :6 $ ! !