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( ) * 22 2 29 2......................................... 2.2........................................ 3 3..................................... 3.2.............................. 3 2 4 2........................................ 4 2.2................................ 4 2.3......................................... 5 2.4..................................... 6 3 6 3..................................... 6 3.2..................................... 7 4 7 4...................................... 8 4.2.................................. 8 4.3.................................. 8 * Emil: hungzh@whu.edu.cn

4.4.................................... 8 4.5............................... 9 5 6 6. ( )........................... 6.2.................................... 6.3............................... 6.4................................ 6.5................................... 2 7 2 8 3 9 3 9........................... 3 9.2........................ 3 4 A............................. 4 A.2.................................. 5. (). (2) (, ); (, ). (3). (4) ; ;. (5). (6) ( ); ;. (7). y + py + qy =, y = f(y, y ), y + P ()y = Q(). (8). (9). 2

.2.2 (),. (2). (3),. (4). (5). (6). :?? :..., y = f(). :,,., ( );,.,,..,.,.,,,..2. ( ) : u() v(). ( ) :. f() d. ( ).,. 3

2 :? 2 : ; ;. 2. f() =, f(). (.) (). (2). ( sin,.) (3) ( ) ( ). (, 3 2,, 3 2.) (4),,. (, 3 3 + 4 2 + 2.),. : 3 3 + 4 2 + 2 7 3 + 2 +, 3 3 + 4 2 + 2 7 3 + 2 +, 3 2 2 2 3 2 + 5, (2n + ) (3n 2) 2, n (5n + 4) 3 n n2 + 2n n n, n 3 n2 sin(n!). n + 2 f() = (cos 4) sin + 3, () df() d( 2 ) ; (2), f()? 2.2, e ln( + ) sin tn rcsin rctn, cos 2 2, ( + ). 4

2.3 2,. sin = 3! 3 + 5! 5 7! 7 +,, sin = +o(), sin. cos = 2! 2 + 4! 4 6! 6 +,, cos, cos = 2! 2 + o( 2 ), cos 2! 2, cos 2 2. 2.3 ( ) + = e., α(), β().. ( ) β() + α() = ep α()β()}, () u() v(),,. u() v() = e v() ln u(),. u()v() = e v() ln u() = ep ( )} v() ln u(). ( + α() ) β() = ep ( β() ln ( + α() ))}, α(), ln ( + α() ) α(), (). (),. 3 : ( + ) 2, 4 ( 2) ( + 2 ( 2 ) ( k, 2 + 3 ) +, ( + 2) 3 sin. 2 + ) =.. ( + 2 2 ) ( ) = + 2 2 5

2.4 3 2 = ep 2 = ep = ep ln 2 2 } 2 ln 2 } 2 } = 2. ( () ) ( ) 5 ( 22) π 4 (tn ) cos sin =.. π 4 (tn ) cos sin ( ) = π + (tn ) cos sin 4 = ep = ep = e 2. π 4 π 4 tn } cos sin sin cos cos cos sin } ( () ) 2.4. 6 : sin, 2 sin, rctn + sin, + 2. :? 3 : ; ;,. 3.,., : f() = f( ). 6

3.2 4, : (i) f( ), ; (ii) f(), ; (iii) =., ;,. 2e, <, 7 f() =, =, 3 +, >. f(),. sin 8 f() =, >, 2 2 +,. f() =,. tf(t) dt 9 F () =,, 2, =, (), F () = ; f(), f() =. (2) F () (, + ) ; F () (, + ). f(), F () f() F (), F () =, F () =. f() =. 3.2,. 4 :?? : ; ; ;. : ;. 7

4. 4 4. : f ( ) = f() f( ). : ;. : ;. (.) : ) f() = sin = ( ) (A) (B) (C) (D) sin 2) f() =,, = ( ), = (A) (B) (C) (D) 4.2. 2 ) f() = 2,,, <, f (). 2) f() =, f (). 3 f() = 2,, + b, >, =,, b. 4.3 : u() v() = e v() ln u(). ( u() >.) 4 : () ; (2) ; (3) ; ( > ) 4.4. 5 dy d : () y = e +y ; (2) cos( 2 + y) = ; (3) y e y = ln 3. 6 u = e y, y = f() y e t2 dt = 2 cos t dt, du d. 8

4.5 4 7 y = f() y = y, dy d2 y d 2. 4.5.. 8 y = y (), d2 y d 2 t=. dy dy dt = dy = dt d d dt ey cos t = ey cos t e y sin t 2 y dy d = y, = 3t 2 + 2t + 3, (2) y = + e y sin t (3). (3) t dy dt = ey dy dt sin t + ey cos t,. d dt = 6t + 2, dy d = e y cos t (2 y)(6t + 2). (4) ln y = y + ln cos t ln(2 y) ln(6t + 2), y y = y + cos t ( sin t) dt d 2 y ( ) y + 6t + 2 6 dt d = y + ( tn 6 ) 6t + 2 6t + 2 + 2 y y. ( dt = ) d 6t+2 t =, y = + e y sin, y =. (4) dy = e d t= 2, = 9 y = t 2 u ln u du, t 2 u 2 ln u du, d 2 y d 2 t= = e2 2 3 4 e. (t > ), d2 y d 2. 9

6 5 :?? : ;,,. ;. ;. 6 :??? : ;. 6. ( ) ( ) f ( φ() ) g() d. :,, φ() g(), φ () = g(), d ( φ() ) = g() d. f ( φ() ) g() d = f ( φ() ) d ( φ() ), φ() u, f(u) du. 2 : sin d, d ln, sin + cos 3 sin cos d, 2 rccos 2 d, + ln ( ln ) 2 d 6.2,,,.,. 2 e d, e d, 2 e d = 2 d(e ) = 2 e e d( 2 )

6.3 6 = 2 e 2e d, 2.,,. 2 e d = 2 e 2e d = 2 e 2 d(e ) = 2 e 2 ( e e d ). 2 : sin d, e sin d, ln d, ln d, e d 6.3 : f() d : d d g() d f() d = f(). d f() d = f ( g() ) g (). 22 : d d f(t 2 ) dt, d tf(t 2 ) dt, d d 2 f(t 2 ) dt, d 2 sin t dt 4, 23 () f(t 2 ) dt = 3, f() d. 2 (2) f(t) dt = 4 4 2 2, f(). 6.4 24 : () 2π sin d; (2) 5 2 3 2, < 2, f() d, f() = + 2, 2.

6.5 7 6.5 25 : f(), g(),, f () g (),, f() f() g() g(). f () g () f () d g () d. f () d f () d, f () d = [ ] f() = f() f(), g () d = [ g() ] = g() g(), f() f() g() g(). :? 7 : ; ; ;. t cos u = u du, 26 t t = t = π sin u y = u du 2. 27,. Y O. y+dy y X 2

9 + 8 d, p < ; p.. p d, p > ; p.. p 9. 9 y + py + qy =, (p, q ) r 2 + pr + q =, r, r 2. () r r 2, y = C e r + C 2 e r2. (2) r = r 2 = r, y = (C + C 2 )e r. (3) r,2 = α ± iβ, y = e α (C cos β + C 2 sin β). 9.2 y + py + qy = f(), (p, q ) (5), y + py + qy =. () f() = e λ P m (), λ, P m () m. (5) y = k Q m ()e λ, k λ r 2 + pr + q =, k =,, 2. Q m () m,. (2) f() = e λ( P l () cos ω + P n () sin ω ), P l () l, P n () n. m = ml, n}. λ ± iω r 2 + pr + q = k, k =,, (5) y = k e λ( Q m () cos ω + R m () sin ω ), Q m (), R m () m,. 3

A.,,?. 28 2 cos π,, f() =, = =, =. =. f () = f() f() = 2 cos π = ( δ, δ), n, n = n+ 2 2n = 2. f( 4n+ 2n) =, f ( f( 2n ) = 2n + ) f( 2n ) = ( 2 = ( 2 ) 2 4n + = ( 2 4n + + ) 2 π cos 4n+ 2 4n+ + cos ) 2 cos (4n+)π 2+(4n+) (4n+)π 2+(4n+) = ( 2 ) 2 4n + sin (4n + )π 2 + (4n + ) 4 = (4n + ) (4n + )2 π 2 4 = π. sin ( π 2 + 2nπ) =, ( δ, δ). (cos ( (4n + ) 2 π ) ( ) 2 2 + (4n + ) (4n+)π 2+(4n+) ) ( ) f +( 2n ) = π. f() 2n. f() 2n+. f() n = n+ 2. 4

A.2 A.2.?. 29 =,. 3 2 sin f() =,,, = 6sin f () = 3cos,,, = ( 6sin 3cos ) 5