II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ = α ± iβ α, β ( x(t) = e αt cos βt α ) β sin βt x + ax + bx = 0 x(0) = 1, x (0) = 0 1
1.5* 5 23 Matlab (1) x t 2 (2) x = 2tx 1 + t 2 (3) x = (1 t)x + t 2 1.6 5 23 (1) x = t + x (2) x + 1 (3) x = t x (4) x 2 t 2 (a) (b) (c) (d) 1.7* 5 23 (1) x = 2(x 4), (0, 1); (0, 4); (0, 5) (2) x (t + x), (0, 1); (0, 2); (0, 1 4 ); ( 1, 1) (3) x 2, (0, 1); (0, 2); (0, 1); (0, 0) (4) x = (x 1)(t + 2), (0, 1); (0, 1); (0, 3); (1, 1) (5) x = tx t 2 +4, (0, 2); (0, 6); ( 2 3, 4) (6) x (2 x), (0, 1 2 ); (0, 1); (0, 2) (7) x = (sin t)(sin x), (0, 2) 1.8 Matlab (1) x = cos(2t x), x(0) = 2 2
1 (2) x 2 ln x, x(0) = 1 3 2.1* 5 23 x 1 = x 2 2.2* 5 23 txx = t 2 + x 2 x(1) = 0 2.3* 5 23 3 (t 2 t)x + (1 2t)x + t 2 = 0 2.4* 5 23 [1] p.7 (1) x = cot t cot x (2) (1 + t)x + (1 x)tx = 0 2.5* 5 23 [1] p.9 (1) x = 2tx t 2 + x 2 (2) t tan x t x + tx = 0 2.6* 5 23 [1] p.10 (1) x = x t + t3 (2) x + x cos t = sin t cos t 2.7* 5 23 [1] p.11 (1) x + x = t sin t, x(0) = 1 3
(2) tx + x = t 3, x(1) = a R 2.8 5 23 [1] p.12 (1) 4tx + 2x = tx 5 (2) 3x x tan t 2 cos t 2.9 5 23 [1] p.13 (1) x 2 x 2 (2) x = x 2 + x t + t2 2.10 R L V I + V switch I R L 2.11 25 NaCl 300 1 0.3 15 /1 12 /1 (1) t NaCl kg/min (2) t l (3) t NaCl kg/min (4) (5) 25 2.12* 5 23 4
(1) tx + x = e t (2) tx + 3x = sin t t 2 (3) x + (tan t)x = cos 2 t, t ( π 2, π 2 ) (4) tx + 2x = 1 1 t (5) (1 + t)x + x = t, t > 0 (6) tx x = 2t log t, t > 0 (7) (sin t)x + (cos t)x = tan t, t (0, π 2 ) 2.13* 5 23 (1) x + 2x = 3, x(0) = 1 (2) tx + 2x = t 3, x(2) = 1 (3) tx + x = sin t, x( π 2 ) = 1 (4) (t + 1)x 2(t 2 + t)x = et t + 1, x(0) = 5 (5) x + tx = t, x(0) = 6 3.1* 5 23 (1) x 3 + 3tx 2 x = 0 (2) x 2 + 3txx = 0 (3) x 4 + 3tx 3 x = 0 3.2* 5 23 [1] p.15 (1) (x sin t t) + (x 2 cos t)x = 0 (2) (1 + t t 2 + x 2 ) + ( 1 + x t 2 + x 2 )x = 0 3.3* 5 23 (1) (sin x sin t) + t cos x x = 0 (2) xe t + (e t cos x)x = 0 5
3.4* 5 23 (1) (1 + tx) + t 2 x = 0 (2) t + (x t)x = 0 (3) (t 2 + 2t x 2 ) 2xx = 0 (4) (t 2 x 2 x) tx = 0 3.5* 5 23 [1] p.16,18 (1) (t + t 2 + x 2 )dt + tx dx = 0 (2) (sin t t cos t 3t 2 (x t) 2 )dt + 3t 2 (x t) 2 dx = 0 (3) x dt + 2t(1 + tx 3 )dx = 0 (4) (x 2 + tx)dt t 2 dx = 0 3.6* 5 23 t 0 = 0 (1) x = t 2 x (2) (t 3)x + 2x = 0 3.7 t 2 x + tx + t 2 x = 0, x(0) = 1, x (0) = 0 0 (1) (2) (1) [ 5, 5] [1] 28-31 A1*, A2*, A3, A4, A5, A6, A7, A8, A10, A11, B1*, B2, B3, B4*, B5, B6, B7, B8, B10, B11, B12, B13, B14 5.1* 6 6 [1] p.63 (1) x + 3x + 2x = 0 6
(2) x + 8x + 12x = 0 (3) x 7x + 12x = 0 5.2* 6 6 [1] p.64 (1) x + x + x = 0 (2) x 3x + 2x 6x = 0 (3) x x = 0 5.3* 6 6 [1] p.68-69 (1) x + x + x = 5t 2 t (2) x 2x + x = 3t 2 + 1 (3) 2x + x x = t 3 (4) x x + x = e 2t (5) x + 2x + x = te t (6) x x = t sin t 5.4* 6 6 (1) f 1 (x), f 2 (x) 2, f 3 (x) = 3x 4x 2 (2) f 1 (x) = 0, f 2 (x), f 3 (x) = e x (3) f 1 (x) = 3, f 2 (x) = cos 2 x, f 3 (x) = sin 2 x (4) f 1 (x) = 1 + x, f 2 (x), f 3 (x) 2 5.5 6 6 (1) t 2 x 7tx + 16x = 0, x 1 (t) = t 4 (2) 4t 2 x + x = 0, x 1 (t) = t ln t (3) (1 2t t 2 )x + 2(1 + t)x 2x = 0, x 1 (t) = t + 1 5.6* 6 6 7
(1) x + 2x + x = 0 (2) x + 3x 4x = 0 (3) x 2x 3x = 4t 5 + 6te 2t (4) x + x = 4t + 10 sin t (5) x + x = 1 t 2 e t (6) x 2x + x = et 1+t 2 5.7 70 cm 0.5 kg 168 cm 40 cm x(t) / 5.8 F sin γt x + ω 2 x = F sin γt, x(0) = 0, x (0) = 0 F γ ω γ ω t 5.9 F γ ω x + ω 2 x = F cos γt, x(0) = 0, x (0) = 0 γ ω ε = 1 2 (γ ω) x(t) = F sin εt sin γt 2εγ Matlab 5.10 (1) x + x + x + x 3 = 0, x(0) = 3, x (0) = 4 (2) x + x + x + x 3 = 0, x(0) = 0, x (0) = 8 (3) x + x + x x 3 = 0, x(0) = 0, x (0) = 1.5 (4) x + x + x x 3 = 0, x(0) = 1, x (0) = 1 8
5.11 Duffing x + x + k 1 x 3 = cos 3 2 t, x(0) = 0, x (0) = 0 k 1 < 0 5.12 d 2 θ dt 2 + 2λdθ dt + ω2 sin θ = 0 λ, ω λ 2 ω 2 5.13 h 0 v 0 h 0 6.1* 6 6 A e ta 5 3 (1) A = 1 1 1 4 2 (2) A = 3 4 0 3 1 3 5 4 2 (3) A = 12 9 4 12 8 3 5 4 (4) A = 2 1 1 1 0 (5) A = 2 3 2 0 2 1 6.2* 6 6 [1] p.94 A e ta (1) x 1 2 1 0 (2) x 1 1 1 1 9
(3) x 0 2 1 2 (4) x 1 0 2 1 0 2 4 (5) x = 2 3 2 x 4 2 0 6.3* 6 6 A x 0 e ta 4 12 5 (1) A =, x 0 = 3 8 1 9 7 3 1 (2) A = 16 12 5, x 0 = 1 8 5 2 1 x = Ax, x(0) 0 7.1* 6 6 6.1-6.3 8.1* 7 11 [1] p.133 (1) x 1 1 3 2 (2) x 1 4 1 1 (3) x 3 1 = x 4 3 (4) x 1 1 1 3 (5) x 1 1 = x 2 1 8.2 dθ dt dv dt = v = sin θ 10
(θ(t), v(t)) 1 2 v2 cos θ 8.3 2 Lotka-Volterra x = (a + by αx)x y = (c + dx βy)y (1) α = β = 0 a, b, c, d (2) α, β > 0 [1] 11