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3 (differential) (equation).. () d/d (). e 0.2 (, ), d/d 0.2e 0.2. e 0.2 (). ()., ().,.. (DE: differential equation). (tpe), (order), (linearit). (ODE: ordinar differential equation). (2). (PDE: partial differential equation). (3).

4 d/d, d 2 /d 2, d 3 /d 3,... Leibniz (Leibniz notation),,,... (prime notation). (2) 5 e 6 0., 3 (4). n d n /d n (n). Leibniz. d 2 /dt 2 6 0 t. Newton (dot notation, flspeck ) t. d 2 s/dt 2 32 s 32. (subscript notation). (3) u u 0 u u tt 2u t.. 2 2. M(, )d N(, )d 0. ( )d 4d 0 d/d, d 4.. n. (4) F n 2,,,,..., (n). (4) (n) n. (4). (5) f (5) (4) (normal form). 2. 4 ( )/4.

5. n (4) F,,, (n) (4) (linear). (4) a n () (n) a n () (n )... a () a 0 () g() 0 (6) n. (6) (n ) 2 (n 2). (7) (6).,,..., (n)..,,..., (n) a 0, a,..., a n., 2, 3. 4. (nonlinear). sin e., 2, 4. 2, (solution).. I I n,,..., (n), n I.

6, n (4) n! I. I.. e 0.2 (, ) d/d 0.2. ()...2 I,,, (a, b), [a, b], (a, ). (, ). (a) d/d /2 ; 4 /6 (b) 2 0; e. (a) (a).. /2 2 /4 4 /6. (b) e e e 2e. (a) 2 (e 2e ) 2(e e ) e 0 (a) 0., 0. 0 I, (trivial solution). (solution curve). I.. I( )..

7 / 0. /..(a) / 0, ( 0) / 0. / 0 ( ). /,. / 0, ( 3, ), ( 2, 0), (, 0), (0, ). 3 2 0 / 0 0 / I. I (, 0) (0, ). (0, ).(b).. (eplicit solution).,, (). 4 /6, e, / d/d /2, 2 0, 0. 0. ().. G(, ) 0. (a) = /, 0 (b) = /, (0, ) / / I G(, ) 0 (4) G(, ) 0 I (implicit solution). G(, ) 0. G(, ) 0 I, G(, ()) 0. G(, ) 0.. 2 2 25 5 5.

8 5 (8) 5 5 2 2 25 5 (a) 2 + 2 = 25 5 (b) 5 5 5 5 (c) 2 = 25 2, 5 < < 5 5 = 25 2, 5 < < 5 (8) 2 sin 5 c > 0 c = 0 c < 0 d/d (8). 2 2 25 'ƒ25 ß 2 () 'ƒ25 ß 2 2 () 'ƒ25 ß 2 2 2 25 2 2 2 25 5 5..2(b) (c).2(a). 2 2 c 0 c (8).. 2 2 25 0.. ( )..,. c. F(,, ) 0, c. c G(,, c) 0 - (one-parameter famil of solutions). n F(,,,..., (n) ) 0 n- G(,, c,..., c n ) 0.., (particular solution). - c cos (, ) 2 sin ( )..3. cos c 0. (, ) c e c 2 e 2 2 0 2- ( ). 0(c c 2 0), e (c 0, c 2 ), 5e 2e (c 5, c 2 2)... t,. c cos 4t c 2 sin 4t. c c 2.

9 c cos 4t t 2 4c sin 4t 6c cos 4t. c 2 sin 4t 6c 2 sin 4t. 2- c cos 4t c 2 sin 4t. (piecewise-defined). - c 4 (, ) 4 0 -..4(a). c = c = (a) c 4 c. 0 c, 0 c.4(b). c =, < 0 c =, 0. (singular solution). 4 /6 0 (, ) d/d /2. 2.2 4 - ( 2 c) 2. c 0 4. 0 ( 2 c) 2. c 0. 4 6 (b) 4 0.. (sstem of differential equations). t,

0 (9). (9) I (9) (pair) (t), 2 (t).. 33 34. (i). 3 2 2 25 (8) () 'ƒ25 ß 2 2 () 'ƒ25 ß 2.. G(, ) 0,,,..3. G(, ) 0, G(, ) 0. G(, ) 0.. 4 42. 2.2 4. (ii).. 35.. (iii) M(, )d N(, )d 0.. 9 0. (iv) F(,,,..., (n) ) 0 (n)... 48 49. (v) I n F(,,,..., (n) ) 0 n- G(,, c,..., c n ) 0 c i, i, 2,..., n (general solution).,.,,,..

. 2.3 3. 8 (6).. ( ) 4 5 cos 2. 3. t 5 (4) t 3 6 0 4. 5. 6. 7. (sin θ) (cos θ) 2 8. I. 5. ( ) 8; 4'ƒ 2 6. 25 2 ; 5 tan 5 7. 2 2 ; /(4 2 ) 8. 2 3 cos ; ( sin ) /2 9 20. ().. I. 9. 20. 2 d ( 2 ) d 0; 2 2 2 9 0 (7). 9. ( 2 )d d 0; ; 0. udv (v uv ue u ) du 0; v ; u 4. I.. 2 0; e /2 2. 3. 6 3 0; e 3 cos 2 4. tan ; (cos ) ln(sec tan ) 2 24. I. 2. 22. 23. 24. 25. 5 8 (). 2 (, ) 2 0

2. 26. 3 () 'ƒ25 ß 2 2 () 'ƒ25 ß 2 ( 5, 5) d/d /. ( 5, 5). 27. e m m. (a) 2 0 (b) 2 7 5 0 28. m m. (a) 2 0 (b) 2 7 5 0 29 32, c 0. 29. 3 5 0 30. 2 2 3 3. ( ) 32. 4 6 0 33 34 (, ). 33. 35.. 36. 0. 37.? k?. 38. 2 ( )? 2? 2. 39. sin d/d 'ƒ ß 2. I.[ : I.] 40. 2 4 5sint Asint B cost(a, B ). Asint Bcost A B. 4 42 d/d f(, ) G(, ) 0. G(, ) 0... I. 4. 34. 4

3 42. F(,,, ) 0. c c 2. 42 43. - 3 3 3c (folia of Descartes).. 44..6 43 c. : 43 3 3 3? I? 42. 45. 3 () 2 () I ( 5, 5). I [ 5, 5]. 46. 2 P P( P) -. (0, 3)? (0, )? 47. d/d f () d 2 /d 2 f (). 48. ( ) 2 4 2 3 0 (4). d/d f (, )? 49. n (4) (5) (5) (4) (equivalent). F(,, ) 0 d/d f (, ). 50. c c 2 2 2-2 (). 5 54 d/d d 2 /d 2. 5. d/d e 2. (a). (b) lim d/d lim d/d? $ $? (c). (d) (a) (c) (). 52. d/d 5. (a) 29 32. (b) (). (). 53. d/d (a-b). a b 0. (a) 29 32. (b) (). (). (c) () a/2b. (d) (a).. (b) (c) (). 54. 2 4. (a). (b) ().? (c) 0.

4 (d) (a) (c) (). 55 56 CAS. 55. (4) 20 58 580 84 0; e 5 cos 2 56. 3 2 2 20 78 0;. ().. 0 I (IVP: initial-value problem). 0,,..., n () n 0, ( 0 ) 0, ( 0 ),..., (n ) ( 0 ) n (initial condition). () n (nth-order initialvalue problem). (2) ( 0, 0 ) I (3) ( 0, 0 ) I 2 m = 2.. (2) 0 I ( 0, 0 )..7. (3) ( 0, 0 )..8. t (t 0 ) 0 (t 0 ) t 0 (sstem).

5 n n- n n. ce (, ) -. (0) 3 0, 3 3 ce 0 c. 3e. (0, 3) (0, 3) (, 2) () 2 2 ce c 2e. 2e. (, 2).9.. () I ( 0 ) 0. I 2.2 6 - /( 2 c) 2 2 0. (0) 0 /c c. /( 2 ). 3. /( 2 ), (),..0(a). /( 2 ) 2 2 0, /( 2 ) I ()..0(a) /( 2 ),,. /( 2 ) 2 2 0, (0), () 0 /( 2 ) I...0(b). (a) = ± (0, ).2 3 6 2. 2. 4 c cos 4t c 2 sin 4t 6 0 2-.. (b) = 0 2

6 (4) ( /2) 2 c cos 2 c 2 sin 2 2 cos 2 sin 2 0 c 2. ( /2) (t) 2 cos 4t c 2 sin 4t. (t) t /2 8 sin 2 4c 2 cos 2 c 2 /4. (4).. (2). e d/d f(, ) ( 0, 0 ) { ( 0, 0 ) 3 (the solution) (a solution). (a)... = 0 = 4 /6 (0, 0) 0 4 /6 d/d /2 (0) 0. d/d /2, (0) 0.. (0, 0).... (2).. 2 3.

7 ( 0, 0 ) a b, c d d R R. f(, ) f/ R a b I 0 : 0 h 0 h, h 0 (2) ( 0, 0 ) (). c a I 0 b f(, ) f/...2. R 3 3 d/d /2 (0, 0) 2. 0 (upper-half plane).. ( 0, 0 ), 0 0 0. d/d /2, (2) 2.., (0) 3, () 2 3e 2e. f(, ) f/. (, ). () (2). 3, (), () I, I 0.. 2 I. ( 0, 0 ). R. R f(, ), I d/d f(, ), ( 0 ) 0. I (), 0, f(, ) ( 0 ) 0..2 3 34. f/ R 0 I 0 ( 0 ) 0.. I I 0. I R I 0 I

8 I 0 : 0 h 0 h h 0 () ( 0, 0 )..2 44. (i).. f(, ) f/ R ( 0, 0 ) (2)... (2),. 4 d/d /2 0., 3 h h (0) 0 2., d/d. d/d, (0).. (ii).2 43,,,. 2 /( c e ) 2 -... (0) 3 2. ( ) 2 3 6 /( 2 c) 2 2 0 -. I. 3. (2) 3 4. ( 2) 2 5. (0) 6. a 2 b 4 7 0 c cost c 2 sint 2 0 2-.. 7. (0), (0) 8 8. ( /0) 0, ( /2) 9. ( /6) 2, ( /6) 0 0. ( /4) '2, ( /4) 2'2 4 c e c 2 e 2 0 2-... (0), (0) 2 2. () 0, () e 3. ( ) 5, ( ) 5 4. (0) 0, (0) 0 5 6 2. 5. 3 2/3, (0) 0 6. 2, (0) 0

9 7 24 ( 0, 0 ). 7. 8. 9. 20. 2. (4 2 ) 2 22. ( 3 ) 2 23. ( 2 2 ) 2 24. ( ) 25 28 'ƒ 2 9.. 25. (, 4) 26. (5, 3) 27. (2, 3) 28. (, ) 29. (a) -., (0) 0. (b) R ( 0, 0 ) R (a). (c) (0) 0. (a). 30. (a) tan ( c) 2 -. (b) f(, ) 2 f/ 2. R. (a) 2, (0) 0. 0 0 2 2. (c) (b) I. 3. (a) /( c) 2 -. (b) f(, ) 2 f/ 2. R. (a) (0) (0).. 32. (a) 3 2, ( 0 ) 0, 0 0. I / 0 / 0. (b) 2, (0) 0 I. 33. (a) 3 2 2 c d/d 3 -. (b) 3 2 2 3. (a) 3 2 2 3. I. (c) ( 2, 3) 3 2 2 3. (b) ( 2) 3. 34. (a) 33 (a) d/d 3, (2) 4. I. (b) d/d 3. 35 38 2 d 2 /d 2 f(,, ).. (a) (), () 2 (b) ( ) 0, ( ) 4 (c) (), () 2 (d) (0), (0) 2 (e) (0), (0) 0 (f) (0) 4, (0) 2

20 35. 5 (0, 9) f(). 40. (, ) 2 2 2 5 5 f(). 4. 2, (0).7 2 5. 35 36. 5 (0, /2) 5 4 37. 5 36 5 42. 2 3 6, ( 0 ) 0 ( 0, 0) 0. 43. d/d f(, ) - f(, ) R 5.. R ( 0, 0 ). 5 44. 37 38. 5 5..8 (a) (b). (, ) d/d /2, (2). 5 5. 38 39 40. 47 (2) (3). 39. (, ) 8e 2 6 (a) (2, ) (b) (2, ) 44

2 45... P t (ear). t 0. 500.... Leonardo da Vinci v gt.,,. 2 3.,,. (mathematical model).... (level of resolution).....,..

22..... t (state of the sstem). t ( ),,. (population growth) 798 Thomas Malthus. Malthusian.*, t. P(t) t,. dp dt r P dp dt kp () k. 790 860. () (). * u v u v u kv

23... Ra-226 Rn-222. (radioactive deca) da/dt t A(t),. (2) () (2).. () (2) k k 0 k 0. () r S ds/dt rs. (2) 50%. (2)... () (2) P 0 A 0. t 0 P(0) P 0 A(0) A 0 3.9. Newton / (Newton s law of cooling/warming). T(t) t, T m, dt/dt, Newton /. (3) k. T m k 0.. (t), (t). d/dt. (t) (t). (4)

24 k. n. (t) (t) n. (4) (4). (5) (5) (0). (2) (first-order reaction).. A. X(t) t A dx/dt kx. X k. t- [(CH 3 ) 3 CCl] t- [(CH 3 ) 3 COH]. t-. (CH 3 Cl) (NaOH) (CH 3 OH) (NaCl).. X A( ) B( ) C( ) X X. C. (6) k. (6) 2 (second-order reaction). 3 300. 300 (gallon). 2 (lb) 3..9. A(t) t ( : lb) A(t). 3 (7)

25 R in,. 300. c(t) A(t)/300 lb/gal.. (7) (8). r in r out * r in r out, r in r out, r in r out 3. (8) r in r out. r in r out r in r out r in r out..3 0 2. Torricelli h v h ( ). v ' 2gåh. g. mv 2 mgh v., t h..20. A h ( : ft 2 ), v ' 2gåh( : ft/s) A h ' 2gåh( : ft 3 /s). t V(t). 2 A w h A h (9) * R in R out.

26 V. E(t) L R. t V(t) A w h dv/dt A w dh/dt. A w ( : ft 2 ) C (a) LRC. dv/dt A w dh/dt (9) t. i di dt L A w (0). h, A w A(h)..3 4. (0) i i R C C (b) i(t) q(t) (A) (C) q.2(a) (inductor), (resistor), (capacitor) (series circuit). t i(t), q(t). L, C, R,,. Kirchhoff 2 E(t)..2(b),,. i(t) q(t) i dq/dt, 3, 2. () () 3.8. v 0 s 0 s(t) Newton 2. Newton. F F k 0 a 0. Newton 2 0 a, F ma. m..22. t s(t)? 2 d 2 s/dt 2. Newton 2.

27 (2), F F W. m, g W mg. (2). s 0 v 0 s 2. (3) (3) g t. (3) s(t) gt 2 v 0 t s 0. 2 Galileo... (3). m v. F F F F 2 mg kv. F, F mg F 2, F 2 kv, (viscous damping)..23. v a a dv/dt Newton 2 F ma mdv/dt. Newton 2, t v(t). kv mg m (4) k. t s(t) v ds/dt a dv/dt d 2 s/dt 2. (4) s 2. (5) L/2 L/2 = 0 L/2 L/2 + L (ft). lb/ft..24(a). (a) = 0 (b) t = 0 (t) (c) t > 0

28. (t) t. 0..24(b) 0 (ft) t 0..24(c). a d 2 /dt 2 ma F (6). (a) (b),,..25(a).25(b).. P P 2..26 P P a (0, a). P P 2 T T 2, P P 2 W. T T, T 2 T 2, W W. T 2 T 2 cos T 2 sin. T 2 T2 sin θ. T 2 tan W/T. P 2 θ T 2 cos θ d/d tan. P T (0, a) w (, 0) (7). 2.2 3.0 (7).

29 t.. (dnamical sstem) (state variable) t 0 ( ).,,..,.. t. t 0. (response of the sstem). da/dt ka. t 0 A(t 0 ) A 0 t t 0 t A(t) A 0 e (t t0) (2.7 ). A(t)., s(0) s 0, s (0) v 0 d 2 s/dt 2 g s(t) gt 2 v 0 t s 0, 0 t T. T. s(t) s (t) t. [t 0, T] t s(t) s (t) v(t) t 0 s (t). s (t) a(t) s (t) g, 0 t T... 2. () r 0 P(t). r 0 P(t)? 2. ()..,. t P(t). 3. 2 t P(t).

30 4. h 0 3. 5. (3) Newton..27 T(t) T m, T 0, k. T 200 50 00 50 5 0 50 00 t 6. (3) T m t. T m (t).28 24. T(t). T m (t) 20 00 80 60 40 20 0 2 24 36 48 t 6 7. 000., (t). 8. t 0 n., t (t). 9. 50 (lb) 300. 3 t A(t). A(0)? 0. 50 (lb) 300., 2 3 2. t A(t).. 0 3.5 A(t)? 2. N 0. r in r out c in c(t) t. t A(t) (8). 3. A h. ca h ' 2gåh. c(0 c )..29 t h. 2 (inch) g 32 ft/s 2.

3 h A w 0 ft 3 4..30. 2, g 32 ft/s 2 3 / c 0.6, t h. 8 ft 7..33 v(t). m v(t). kv 2 mg A w 20 ft h 7 4 5..3. R, L E(t), i(t). L E R 5 LR 6..32. R, C E(t), q(t). R E C 6 RC 8..34(a) s (ft) w (lb)...34(b) (t). Archimedes :,. 62.4 lb/ft 3. s/2 0 (a) surface s/2 0 (b) (t) 8 9. m.35(b) s. / (t)

32..35(c) /.. Hooke t (t)... (t) 5 lb 2 (a) s (t) < 0 m = 0 (t) > 0 (b) 9 / m (c) 22..37 L (ft). 2 lb/ft. t 0. t (t). (8) 2L (t) 20. 9 /,, (t).. m Newton 2 : 0 F.. L (8) mv. 2 22 Newton 2. 2. 0 (ft)..36 5 (lb). lb/ft. t (t) * m F ma. 22 23. Newton (Newton s law of universal gravitation).38 a g. a r, a k/r 2. k. r R a g k. Newton 2

33 r. m r C S.. C. R (0, s) (, ) M 23 s P θ C 24..39 m. t r, M, M r,,, r,,. m r R 24 25.. M A(t) t. A(t). 26. 25 t. t A(t). 27. r g t (t). t (t). 28. (tractri) P.40, (0, s) 28 29..4 C L O( ). C. C,,.[ : 2.?.] C P(, ) θ L θ φ O 29 30.. 37 () P(t). () -. 3. (3) T m. (3) k 0. T(t) T m. 32. (8). 50 (lb) t 0

34 A(t).. dp 33. dt (k cos t)p P(t) k 0.,. 34..42(a). S. S ( ) S. S C f(). P(, ) m..42(b). (a) P S F. Newton 2 m 2.?.42(b). (a) (b) (a) f(). ω P C 35. 23 r R s. s (b) 34. s R 23? θ C P m ω 2 P(, ) θ mg 36. (virga).. t 0 r 0. (a),.. t r(t).[ :. 47.] (b) t v..[ : 2 22.] 37. (snowplow problem) Ralph Palmer Agnew. McGraw-Hill Ralph Palmer Agnew Differential Equations. 38.. 39. P (t) 0.5 P(t). P(t) t( : ). 2 t. 40. 226

35. A(t) t( : ) g. 0.002 g t? 2 c d/d f(,). c k.. 2. 3 4 c c 2 F(, ) 0 2. c, c 2, k. 3. 4. 5 6, c c 2 F(,, ) 0 2. c c 2. 5. c e c 2 e 6. c e cos c 2 e sin 7 2 (a) (d). (a) 0, (b) 2, (c) 2, (d) 2 2 7. 2 8. 2 9. 2 4 0.. 9 8 2. 0 3 4. 3. 4. ( 3) 5 6. 5. () P(, ) P(,). 6. () P(, ) P(, ). 7. (a) 2/3. (b) 2/3 3 2 0 I. 8. (a) - 2 2 2 c (2 2) 2. (b) (a) - (0). (c) (b) (0) ().. ()? I. 2 9. 2. () 0 I. 20. () 2 2, () 2. () (). 2.. (a) () 2 c 2 () 2 c 2. (b) () 2 () ( ) 2 4 2. (c) (b) (a).

36 22. 6' 5 3 (, 4). 5 23 26.. 23. 2 cos 2 sin ; sin cos 24. sec ; sin (cos ) ln(cos ) 25. 2 0; sin(ln ) 26. 2 sec(ln ); cos(ln ) ln (cos(ln )) (ln ) sin(ln ) 27. 2 d 2 /d 2 f(,, ), (2) 0, (2).43. 0. 28. 2, 0. 5 5 27 /2 t h..