- 2011, http://c-faculty.chuo-u.ac.jp/ nishioka/ 2 11 21131 : 4 1 x y(x, y (x,y (x,,y (n, (1.1 F (x, y, y,y,,y (n =0. (1.1 n. (1.1 y(x. y(x (1.1. 1 1 1 1.1... 2 1.2... 9 1.3 1... 26 2 2 34 2.1,... 35 2.2 (2.2... 37 2.3 2... 39 2.4... 42 2.5... 46 2.6... 50 1.1 (1.2 y (x =f(x. y(x =, C x x 0 f(t dt + C. : x = x 0 y(x 0 y 0. 2
1.1 (.. t v(t (1.3 v (t =9.8 ( a. a. 1.2. 0. T v(t u(t. [ ] 20m. 20 20 = 4.9 T 2 T = =2.020 2 4.9 2, ( 20 v 4.9 =9.8 20 = 19.799 20 m/sec = 72km/h. 4.9 72km/h 20m. 3 5 [ ] (1.3. 0, T v(t v(t = T 0 9.8 dt = 9.8T m/sec.,, T u(t u(t = T v(t dt = T 0 0 9.8 t dt =4.9 T 2 m. 1.3. 0 θ. v 0,.. 4 6
[ ]. y y(t x x(t, y (t = x (t = t 0 t 0 y (t = 9.8, y(0 = v 0 sin θ, x (t =0, x(0 = v 0 cos θ 9.8 ds + v 0 sin θ = 9.8t + v 0 sin θ y(t = t 0 0 ds + v 0 cos θ = v 0 cos θ x(t = t 0 y (s ds = 4.9t 2 + v 0 t sin θ x (s ds = v 0 t cos θ 7 1.2. (1.4 y (x =f(x g(y(x 1.4. (1.4. 1 g(y dy = f(x dx [ ] (1.4 g(x. 1 f(x dx = g(y(x y (x dx =,. 9 1 g(y dy. t, 7000 6000 4.9 x2 y = + x tan θ. (v 0 cos θ 2 (. 1.5 ( - 14. t W (t 14. t t + t, 14 W (t W (t + t W (t (. 5000 4000 3000 2000 1000 5000 10 000 15 000 Step 1. (! W (t. (1.5 W (t W (t. (1.6 t, W (t.. λ, : θ = 15, : θ = 30, : θ = 45, : θ = 60, : θ = 75. (1.7 W (t =W (t + t W (t = λ W (t t 8 10
t, t 0 : Step 1. : (1.8 (1.5, (1.6 W W (t + t W (t (t = lim = λ W (t. t 0 t W (t = λ W (t, λ > 0. W (t Step 2. (1.8 W (t : W (t = λ.. c, log W (t W (t = dt = λ dt = λ t + c. (1.9 W (t W (t =e λt+c = e λt e c = Ae λ t. (A e c., g kv 2 (t = ( g + kv(t ( g kv(t. V (t g kv 2 (t = = 1 2 g V (t ( g + kv(t ( g kv(t { V (t + g + kv(t V (t } g kv(t 11 13 1.6.,.. I., : mv (t =mg kv 2 (t,, m =1. g =9.8m 2 /sec, k =. (1.10 V (t =g kv 2 (t, k =. Step 2. (1.10 g kv 2 (t,. 1 2 gk = V (t = { log g + kv(t log g kv(t } V (t g kv 2 (t dt = g k g =9.8m/sec 2, dt = t + c. exp{ gk t} 1 exp{ gk t} +1. k. 12 14
II. (.. (, (, m/sec (, sec (., (. mv (t =mg rv(t, g =9.8m/sec 2,r= 14 12 10 8 6 4 2 0.5 1.0 1.5 2.0 2.5 1.1 r =2., 5m/sec. 15 17 g rv(t, : 1 r log g rv(t V (t = g rv(t dt = dt = t + c. g rv(t = exp{ rt c} V (t = 1 r g ( 1 exp{ rt} V (0 = 0. 1.7 (. Step 1. (i r A. t (1 + r t A. (ii r/12 A. t (1 + r/12 12t A. (iii r/365 A. t (1 + r 365 365 t A. (iv n, r/n A. t (1 + r n nt. (v n ( r (1.11 lim n (1 + r n nt A = e rt A. 16 18
(vi t V (t. (1.11 V (t (! : V (t =re rt A = rv(t. [ ] t t + t, A(t + t A(t = { e r t 1 } A(t K t. t, t 0. A(t + t A(t A (t, t e r t 1 r, t : (1.12 A (t =ra(t K = r { A(t K r }. 1.9. (i (1.12. A(0 = B. 19 (ii T. 21 1.8. r B. 1 K.? 1.8, t A(t. A(t. : r. t V, t, e r t V., t t + t e r t V V = { e r t 1 } V t K t (1 K. [ 1.9 ] (i (1.12 A(t K r : A (t A(t K/r = r. f (x ( : f(x dx = log f(x + C, log A(t K r A(0 = B, t =0, = rt+ c. A(t = K r + Cert, C = e c. B = A(0 = K r + C C = B K r. (1.13 A(t = K r + ( B K r e rt. 20 22
(ii T (, ρ, (1.14 r (1.14 0=A(T = K r + ( B K e rt. r e rt K/r = K/r B = K K Br. T = 1 r log K K Br. r log(1 + ρ., 0.01 ρ 0.09, ρ ( log(1 + ρ (. 0.08 : B K r < 0 K>Br,. 0.06 ( 1000, 100, r 0.01 r 0.09? (1.14 B = 1000, K= 100, T = 1 r log 1 1 10 r. 23 0.04 0.02 0.02 0.04 0.06 0.08 25 25 20 15 10 5 1.3 1 1 x P (x,q(x, (1.15 y (x+p (x y(x =Q(x. 0.02 0.04 0.06 0.08 r = 1% 10.6, r = 4% 12.8, r = 8% 20.1. 1.10. ( y(x exp{ P (x dx}. 24 26
[ 1.10 ] ( exp{ P (x dx} = P (x exp{ P (x dx}. 1 1.12. (1.15 y (x+p (x y(x =Q(x ( y(x exp{ P (x dx} ( = y (x exp{ P (x dx} + y(x exp{ P (x dx} = y (x exp{ P (x dx} + y(x P (x exp{ P (x dx} ( = exp{ P (x dx} y (x+p(x y(x.. [ 1.12 ] 1.11 exp{ P (x dx} ( y(x exp{ = y (x+p (x y(x =Q(x ( y(x exp{ P (x dx} = exp{ P (x dx} P (x} Q(x. 27 29 1.11. exp{ P (x dx} ( y(x exp{ [ 1.11 ] 1.10 ( exp{ P (x dx} y(x exp{ = exp{ P (x dx} exp{ P (x dx} = y (x+p (x y(x. P (x dx} P (x dx}. ( y (x+p(x y(x y(x exp{ = dx ( exp{ ( P (x dx} = y(x exp{ P (x} Q(x + C : y(x =C exp{ P (x dx} + + exp{ P (x dx} dx ( exp{ P (x dx} dx P (x dx} Q(x. 28 30
1 1.13. x P (x,q(x, 1 y (x+p (x y(x =Q(x, : y(x =C exp{ P (x dx} + ( + exp{ P (x dx} exp{ P (x dx} Q(x dx. 1.14. a.. y (t+ay(t = cos t.. 2.0 1.5 1.0 0.5 0.5 2 4 6 8 10 12 14 ay(t (mass dumper, cos t. y(0 = 12/5, a=2 y(t =2e 2x sin t + 2 cos t + =2e 2x + 1 cos(t α, α 0.463 5 5 31 33 [ 1.14 ] y (t+ay(t =e at( e at y(t = cos t. ( e at y(t = e at cos t. e at y(t =C + e at cos t dt 2 2 2. (2.1 y (t =F (y (t,y(t,t. u(t y (t, (2.1 1 : { y (t =u(t, u (t =F (u(t,y(t,t at sin t + a cos t = C + e 1+a 2. y(t =Ce at + 32 sin t + a cos t 1+a 2. 2 (2.1. F,. : b, c (2.2 y (t+by (t+cy(t =0 34
2.1, 2.1. y 1 (t, y 2 (t (2.2. c 1,c 2, (2.2. [ ] : Y (t c 1 y 1 (t+c 2 y 2 (t Y (t+by (t+cy(t =`c 1 y 1 (t+c 2 y 2 (t + +b`c 1 y 1(t+c 2 y 2(t + c`c 1 y 1 (t+c 2 y 2 (t = c 1`y 1 (t+by 1(t+cy 1 (t + c 2`y 2 (t+by 2(t+cy 2 (t = c 1 0+c 2 0=0. 35 2.2 (2.2 (2.3 k y(t =e kt. (2.2 : k 2 e kt + bke kt + ce kt = e kt( k 2 + bk+ c =0. 2.3. (2.4 k 2 + bk+ c =0 (2.2. 37, 2.2. f, g C, f(t =Cg(t, t (2.4 : (2.5 k 1 b b 2 4c 2, k 2 b + b 2 4c. 2, f g. (ii f, g,. Y (t, (2.2. (2.2 3 : 1. (2.4 k 1,k 2, i. e. b 2 4c >0. 2. k 1,k 2, i. e. b 2 4c <0. 3. k 1 = k 2, i. e. b 2 4c =0., α, β Y (0 = α, Y (0 = β. 36 38
2.3 2 k 1,k 2 2. (2.2. k 2 + bk+ c =0 y 1 (t e k 1 t, y 2 (t e k 2 t Y (t c 1 e k 1 t + c 2 e k 2 t, c 1,c 2. 50 000 40 000 30 000 20 000 10 000 1 2 3 4 5 6 39 41 2.4. y (t+y (t 2y(t =0, y(0 = 0, y (0 = 1. 0=k 2 + k 2=(k 1(k + 2 k 1 = 1, k 2 =2, Y (t c 1 e t + c 2 e 2 t 0=Y (0 = c 1 + c 2, 1=Y (0 = c 1 +2c 2 c 1,c 2. c 1 = 1 3, c 2 = 1 3 Y (t =1 3 40 ( e t + e 2 t. 2.4 k 1,k 2 k 2 + bk+ c =0 2. y 1 (t e k 1 t, y 2 (t e k 2 t (2.2. Y (t c 1 e k 1 t + c 2 e k 2 t (2.2., Y (t,., p b 2, q 4c b 2, k 1 = p qi, k 2 = p + qi. k 1, e k 1 t? 42 2.
( II e k 1 t = e (p qit = e pt e iqt = e pt( cos qt i sin qt, e k 2 t = e (p+qit = e pt e iqt = e pt( cos qt+ i sin qt. (2.2 y 1 (t ek 1 t + e k 2 t 2 y 2 (t ek 1 t e k 2 t 2 i = e pt cos q t, = e pt sin q t. (2.2 (!. (2.2 Y (t e pt( c 1 cos qt+ c 2 sin qt. c 1 =0, c 2 = 1 2 Y (t =1 2 e t sin 2t. 0.25 0.20 0.15 0.10 0.05 1 2 3 4 5 6 0.05 43 45 2.5. y (t+2y (t+5y(t =0, y(0 = 0, y (0 = 1. 0=k 2 +2k +5= ( k ( 1+2i ( k +( 1 2 i p = 1, q=2. Y (t e t( c 1 cos 2t + c 2 sin 2t, c 1,c 2. 0=Y (0 = c 1, 1=Y (0 = e t( c 1 cos 2t + c 2 sin 2t +e t( 2c 1 sin 2t +2c 2 cos 2t 44 t=0 t=0 2.5, k 2 + bk+ c =0 b 2 4c =0 k 1 = k 2 = b 2 k. y 1 (t e kt (2.2, y 2 (t. 2.6. b 2 4c =0, (2.2. y 2 (t te kt 46
[ ] y 2 (t (2.2 : y 2 (t+by 2(t+cy 2 (t = ( 2k + k 2 t e kt + b ( 1+kt e kt + c te kt = e kt( (k 2 + bk+ c t +2k + b = e kt( 2k + b = e kt( 2 b 2 + b =0. y 1 (t,y 2 (t. 0.35 0.30 0.25 0.20 0.15 0.10 0.05 Y (t c 1 e kt + c 2 te kt 1 2 3 4 5 6, c 1,c 2. 47 49 2.7. y (t+2y (t+y(t =0, y(0 = 0, y (0 = 1. 0=k 2 +2k +1= ( k +1 2 k = 1. Y (t e t( c 1 + c 2 t, c 1,c 2. 0=Y (0 = c 1, 1=Y (0 = e t( c 1 + c 2 t t=0 + e t c 2 t=0 = c 1 + c 2. 2.6 2.8.. (i y (t+y (t 2y(t =0, y(0 = 0, y (0 = 2. (ii y (t+y(t =0, y(0 = 10, y (0 = 5. (iii y (t+4y (t + 20y(t =0, y(0 = 0, y (0 = 3. (iv y (t+6y (t+9y(t =0, y(0 = 6, y (0 = 4. 2.9.. (i y (t+2y (t y (t 2y(t =0. (ii y (t+3y (t+3y (t+y(t =0. c 1 =0, c 2 =1 Y (t =e t t. 48 50
2.10. b, c > 0, y (t+by (t+cy(t =0 y(t, lim t y(t =0. 51