Enoncé AD N°2 - CPGE Brizeux

!"#$%&'()*+,% -.%/01% -23456*'78*9)(*9%1:;<71:;=%
%
!""#$%&'()$%*+',-./#'(012(34(#',)#'(5'(+'#%#')/(678669(
:-/5/;.-/$,()'(#.)/$,*%5</)';(',(+<)'%/,'(
%
>?@*A3(B%C%D*%?+3%E*%A*33*%5FF'GAH*%EGA+4*935('*%*I3%EJ+3(2(I*'%2*I%AG995(II59A*I%5FFG'3K*I%F5'%2*%AG+'I%E*%
A(9K3(L+*% 45A'GIAGF(L+*% FG+'% 5?G'E*'% 2*I% F'G?2K453(L+*I% 2(K*I% M% 2J+3(2(I53(G9%E*I% '5E(G9+A2K(E*I% *9%
4KE*A(9*%5(9I(%L+JM%25%N*I3(G9%E*I%EKAH*3I%'5E(G5A3(BIO%
%
=#.>./5(4(#<.5/;'#(
? PG93'*'% L+*% 2*I% EG99K*I% *,FK'(4*9352*I% IG93% *9% 5AAG'E% 5Q*A% +9*% 2G(% A(9K3(L+*% ERG'E'*% ;% FG+'% 25%
'K5A3(G9%E*%EKI(93KN'53(G9%E*I%3'5A*+'I%'5E(G5A3(BIO%
? .KN5N*'%2*I%5Q5935N*I%2(KI%M%2R+3(2(I53(G9%E*%2R(GE*%;1S%F5'%'5FFG'3%M%2R(GE*%;S;%C%
• E+%FG(93%E*%Q+*%E+%F53(*93%AG9A*'9K%F5'%+9*%IA(93(N'5FH(*%3H6'GTE(*99*%U%
• E+%FG(93%E*%Q+*%E*%25%N*I3(G9%E*I%EKAH*3I%F5'%2RHVF(352O%
>9%IR5FF+(*'5%I+'%E*I%)<>'5$""'+',-;(,*+<#/@*';(#/A$*#'*B%FG+'%5'N+4*93*'%25%'KFG9I*O%
? WK52(I*'%+9%E(5N9GI3(A%4KE(A52%E*%25%IA(93(N'5FH(*%3H6'GTE(*99*%F'KI*93K*%E59I%2*%EGA+4*93%1O(
(
C$%*+',-(6(D(!%-/>/-<()E*,(<%&.,-/55$,(#.)/$.%-/F(
DR5A3(Q(3K%𝐴𝑡%ER+9*% I+?I359A*% '5E(G5A3(Q*% 4*I+'*%2*% 9G4?'*% 4G6*9% E*% EKI(93KN'53(G9I% F5'% +9(3K% E*%
3*4FIO% X22*% IR*,F'(4*% *9% &*AL+*'*2% YI64?G2*% &LZ% G[% ;% &L% AG''*IFG9E% M% ;% EKI(93KN'53(G9% F5'% I*AG9E*O%
DR5A3(Q(3K%𝐴𝑡*I3%'*2(K*%5+%9G4?'*%E*%9G65+,%E*%2RKAH593(22G9%𝑁(𝑡)%F5'%25%'*253(G9%C%%
𝐴𝑡=−
𝑑𝑁 𝑡
𝑑𝑡 =𝜆×𝑁(𝑡)%
𝜆%*I3%25%AG9I3593*%'5E(G5A3(Q*\%*22*%IR*,F'(4*%*9%I];O%
%
C$%*+',-(2(D(:,('B.+',(+<)/%.5G(5.(;%/,-/A#."&/'(-&H#$I)/',,'(!"#$%&'()*$+),--.)/012342)56)777895"8123:;
&5225(<8=&)56)>66%?@@7778+>;&41A$3B8=&C(
D5% 3H6'GTE*% *I3% +9*% N259E*\% I(3+K*% E59I% 25% 'KN(G9% A*'Q(A52*% 593K'(*+'*\% 5FF2(L+K*% AG93'*% 2*%
25'69,% *3% 25% F5'3(*% I+FK'(*+'*% E*% 25% 3'5AHK*O% D5% BG9A3(G9% F'(9A(F52*% E*% A*33*% N259E*% *I3% 25%
IKA'K3(G9%E*I%HG'4G9*I%3H6'GTE(*99*I%M%F5'3('%E*%2J(GE*%52(4*935('*%L+(%I*%B(,*%3*4FG'5('*4*93%I+'%A*33*%
N259E*O% .*% F*3(3*% 35(22*\% F*I593% ;^% M% 1^% N% AH*)% 2J5E+23*% *3% 4*I+'593% *9Q('G9% _% A4% *9% 25'N*+'% *3% S% A4% *9%
H5+3*+'\%*22*%9J*I3%9G'452*4*93%F5I\%G+%M%F*(9*\%F52F5?2*O%P5(I%*22*%F*+3%IJH6F*'3'GFH(*'\%IG(3%E*%459(`'*%
F2+I%G+%4G(9I%E(BB+I*%*3%HG4GN`9*\%IG(3%E*%459(`'*%2GA52(IK*%5Q*A%25%BG'453(G9%E*%9GE+2*YIZO%"*I%9GE+2*I%
F*+Q*93%F'(9A(F52*4*93%a3'*%E*%E*+,%IG'3*I%C%H6FGB(,593%G+%H6F*'B(,593O%$2I%IG93%E(3I%H6FGB(,593I%IJ(2I%B(,*93%
F*+%EJ(GE*%F5'%'5FFG'3%5+%'*I3*%E*%25%3H6'GTE*O%$9Q*'I*4*93\%(2I%IG93%E(3I%H6F*'B(,593I%IJ(2I%B(,*93%F2+I%EJ(GE*%
L+*%2*%'*I3*%E*%25%3H6'GTE*O%
D5% IA(93(N'5FH(*% 3H6'GTE(*99*% *I3% +9*% 3*AH9(L+*% EJ*,F2G'53(G9% FH6I(G2GN(L+*\% *22*% AG9I3(3+*% +9% *,54*9%
AG4F2K4*935('*%M%2R*,F2G'53(G9%5953G4(L+*%F5'%KAHGN'5FH(*%F5'%*,*4F2*O%"*3%*,54*9%F*'4*3%E*%EKA*2*'%
E*I%9GE+2*I%E*%25%3H6'GTE*O%DG'I%ER+9*%IA(93(N'5FH(*\%+9*%(45N*%E*%2JG'N59*%K3+E(K%*I3%'*AG9I3(3+K*O%%
%
$45N*%G?3*9+*%5F'`I%IA(93(N'5FH(*%E*%25%3H6'GTE*%ER+9%F53(*93%4525E*%C%
%
(
C$%*+',-(J(D(:-/5/;.-/$,()'(-#.%'*#;(#.)/$.%-/F;(!"#$%&'()>66%?@@7778525&D56>3E158+49)56)A$+)/012342),--.C(
!G+'%*BB*A3+*'%+9*%IA(93(N'5FH(*\%G9%+3(2(I*%E*I%3'5A*+'I%'5E(G5A3(BI\%(9@*A3KI%E59I%2*%AG'FI%E+%F53(*93O%X9%I*%
EKI(93KN'593\% 2*I% 3'5A*+'I% K4*33*93% E*I% '56G9I% N5445\% L+(%F*+Q*93% a3'*% EK3*A3KI% F5'% +9% 5FF5'*(2% E*%
4*I+'*% 5FF*2K% bEK3*A3*+'% M% IA(93(2253(G9IbO% "*2+(7A(% '*AG9I3(3+*% +9*% (45N*% E*% 2RG'N59*% M% F5'3('% E*I%
'56G99*4*93I%K4(IO%!G+'%'K52(I*'%+9*%IA(93(N'5FH(*%E*%25%3H6'GTE*\%G9%+3(2(I*%2*I%'5E(G9+A2K(E*I%I+(Q593I%C%
3*AH9K3(+4%cc\%(GE*%;S;\%G+%(GE*%;1SO%

!"#$%&'()*+,% -.%/01% -23456*'78*9)(*9%1:;<71:;=%
%
DG'I%E*%I5%EKI(93KN'53(G9\%2R(GE*%;1S%K4*3%+9%'56G99*4*93%N5445\%E*%2RG'E'*%E*%25%A*935(9*%E*%d*e\%3'`I%
B5QG'5?2*%M%25%EK3*A3(G9\%*3%(2%5%+9%3*4FI%E*%E*4(7Q(*%YG+%FK'(GE*%'5E(G5A3(Q*Z%AG+'3\%E*%;S%H*+'*IO%$2%I*'5%
EG9A%F'(Q(2KN(K%FG+'%2J(45N*'(*\%452N'K%IG9%AGf3%K2*QK%2(K%M%I5%F'GE+A3(G9%F5'%A6A2G3'G9O%
DR(GE*% ;S;% I+?(3% +9*% EKI(93KN'53(G9% EJKL+53(G9% C 𝐼
!"
!"! → 𝑋𝑒∗
!"
!"! + 𝑒
!!
!O% DRK4(II(G9% E*% '56G9I% N5445%
F'GQ(*93%E*%25%EKI*,A(353(G9%E+%9G65+%E*%,K9G9%BG'4KO%D5%FK'(GE*%'5E(G5A3(Q*%E*%2R(GE*%;S;%E*%g%@G+'I%*3%
2JK4(II(G9% ?a35% (4FG'3593*% 2*% '*9E*93% 3'`I% B5QG'5?2*% M% +9*% +3(2(I53(G9% 3HK'5F*+3(L+*O% .*% F2+I\% IG9%
'56G99*4*93% N5445% 3'`I% K9*'NK3(L+*% YF2+I(*+'I% A*935(9*I% E*% d*eZ% B5(3% L+*% 2JG9% F*+3% 2*% AHG(I('% FG+'%
A*'35(9*I%5FF2(A53(G9I%EJ(45N*'(*\%FG+'%A*'35(9I%G'N59*I%EG93%2*%4K35?G2(I4*%*I3%2*93%Y25%N259E*%I+''K952*\%
F5'%*,*4F2*ZO%
;S;I%
;1SI%
cchA%
𝜆;S;%i%;\:1Sj;:k<%Ik;%
𝜆;1S%i%;\_^cj;:k^%Ik;%
𝜆cc%i%S\1;j;:k^%Ik;%
F42(6$265()&$"34$+63:5()"5)E15GE15()6&$+51&()&$"34$+63=()
PGE*%GFK'53G('*%*3%2*A3+'*%ER+9*%IA(93(N'5FH(*%E*%25%3H6'GTE*%C%
>9% (9@*A3*% F5'% *,*4F2*% 5+% F53(*93\% +9*% EGI*% E*% ;1SI\% EJ5A3(Q(3K% -% i% =% P&L\% AG93*9+% E59I% +9*% IG2+3(G9%
EJ(GE+'*% E*% IGE(+4% /5I%G[% 2J(GE*% *I3% 2*% 3'5A*+'% '5E(G5A3(BO% >9% 25(II*% 52G'I% 2J(GE*% I*% B(,*'% I+'% 25% 3H6'GTE*%
F*9E593% L+*2L+*I% H*+'*I\% F+(I% G9% 'K52(I*% 25% IA(93(N'5FH(*O% DR(45N*% G?3*9+*% F'KI*93*% E*I%)G9*I% BG9AK*I%
'*F'KI*93593%2*I%)G9*I%E*%2JG'N59*%BG'3*4*93%K4*33'(A*I%*9%'56G9I%N5445O%>9%AG9I(E`'*%L+*%2*%F53(*93%9J5%
F2+I%E*%3'5A*+'I%E59I%2*%AG'FI%L+59E%(2%5%K2(4(9K%ccl%E*%25%L+593(3K%E*%3'5A*+'I%(9(3(52*O%
(
C$%*+',-( K(D( C<%&'-;( &$;"/-.5/'#;( #.)/$.%-/F;(!"#$%&'() 7778+&33&$"84&D@&$H42259526(@I<;5==G1526(;
>4(%36$G35&(8%"=C%
m9*%N'59E*%F5'3(*%E*%2R5A3(Q(3K%5E4(9(I3'K*%*I3%K2(4(9K*%F5'%2*I%QG(*I%953+'*22*I%E+'593%2*I%H*+'*I%G+% 2*I%
F'*4(*'I%@G+'I% YF5'BG(I% 2*I% I*45(9*IZ% I+(Q593% 2R5E4(9(I3'53(G9% E*I% I+?I359A*I% '5E(G5A3(Q*IO% >9% *I3(4*% F5'%
*,*4F2*% L+R*9Q('G9% g_% l% E*% 2R(GE*% ;S;% IG93% K2(4(9KI% Q(5% 2*I% +'(9*I% E*I% ^% F'*4(*'I% @G+'IO% m9*% F5'3(*%
I*+2*4*93% E*% A*I% *BB2+*93I% *I3% AG22*A3K*% E*% 459(`'*% IFKA(B(L+*% 5+% 9(Q*5+% E*% 2RHVF(352O% "R*I3% 2*% A5I% F5'%
*,*4F2*%F*9E593%25% FH5I*% E*%AG9B(9*4*93%*9%AH54?'*%E(3*%n%F2G4?K*%o% FG+'%2*I%F53(*93I%L+(%I+?(II*93%
+9*%3HK'5F(*%L+(%4*3%*9%@*+%F2+I(*+'I%A*935(9*I%E*%P&L%ER(GE*%;S;O%.59I%A*%A5I\%2*I%*BB2+*93I%IG93%4(I%*9%
533*93*% 5Q593% '*@*3% E59I% E*I% A+Q*I% E*% EKA'G(II59A*O% .59I% 2*% A5I% E*% 2J(GE*% ;S;\% +9% 2(L+(E*% YE*I% +'(9*I% E*%
F53(*93Z%EG93%2*%9(Q*5+%E*%AG9354(953(G9%*I3%E*%;%4(22(G9%E*%&LpD%5+'5%5(9I(%+9*%'5E(G5A3(Q(3K%'KI(E+*22*%E*%
^::%:::%&LpD%5+%?G+3%E*%g%@G+'I\%1^:%:::%&LpD%5+%?G+3%E*%;<%@G+'I%Y1%FK'(GE*IZ%*3%^%:::%&LpD%5+%?G+3%E*%1%
4G(I%Y=\^%FK'(GE*IZO%
%
-%3(3'*%(9E(A53(B\%2R5''a3K%E+%S:%GA3G?'*%;cg;%F'KQG(3%L+*%n%2RKQ5A+53(G9%E*I%A+Q*I%E*%I3GAd5N*%E*I%*BB2+*93I%
2(L+(E*I%9*%F*+3%(93*'Q*9('%L+*%I(%2R5A3(Q(3K%QG2+4(L+*%*I3%(9BK'(*+'*%M%=%?*AL+*'*2I%F5'%2(3'*%oO%D5%L+*I3(G9%
*I3%52G'I%A*22*%E*%25%A5F5A(3K%E*%I3GAd5N*%E*I%A+Q*I%E*%2RHVF(352O%%%
%
C$%*+',-(%$+"5<+',-./#'(D%-9(453(G9%F'KI*93593%2R*,54*9%4KE(A52%C%
H33FCppA5*(9BGO(91FSOB'p$PqpB25IHp59(4Ip5FF4*E(pIA(93(p59(4#A(93(N'5FH(*r(952OIsB%
:%
1:::::%
_:::::%
<:::::%
g:::::%
;::::::%
;1:::::%
:% ^% ;:% ;^% 1:% 1^% S:% S^%
!%L>/-<(".#(5/-#'()M'N*',-;(3',(O@8P9(
='+";(3',(Q$*#;9(
C<%#$/;;.,%'(#.)/$.%L>'()'(6J6I