( ) 1 / 21
1 2 3 4 ( ) 2 / 21
x(t)? ẋ + 5x = 0 ( ) 3 / 21
x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 ( ) 3 / 21
x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 x(t) = sin 5t? ẋ = 5 cos 5t, ẋ + 5x = 5 cos 5t + 5 sin 5t 0 ( ) 3 / 21
x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 x(t) = sin 5t? ẋ = 5 cos 5t, ẋ + 5x = 5 cos 5t + 5 sin 5t 0 x(t) = e 5t? ẋ = 5e 5t, ẋ + 5x = 5e 5t + 5e 5t = 0 ( ) 3 / 21
x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 x(t) = sin 5t? ẋ = 5 cos 5t, ẋ + 5x = 5 cos 5t + 5 sin 5t 0 x(t) = e 5t? ẋ = 5e 5t, ẋ + 5x = 5e 5t + 5e 5t = 0 x(t) = 3e 5t? ẋ = 15e 5t, ẋ + 5x = 15e 5t + 15e 5t = 0 ( ) 3 / 21
x(t)? ẋ + 5x = 0 x(t) = Ce 5t (C ) ( ) 4 / 21
x(t)? ẋ + 5x = 0 x(0) = 2 ( ) 5 / 21
x(t)? ẋ + 5x = 0 x(0) = 2 x(t) = Ce 5t x(0) = Ce 5 0 = C = 2 x(t) = 2e 5t ( ) 5 / 21
x(t)? ẋ + 5x = 0 ( ) 6 / 21
ẋ + 5x = 0 x(t)? x(t) = e λt ẋ = λe λt λe λt + 5e λt = 0 λ + 5 = 0 λ = 5 x(t) = e 5t ( ) 6 / 21
ẋ + 5x = 0 x(t)? x(t) = e λt ẋ = λe λt λe λt + 5e λt = 0 λ + 5 = 0 λ = 5 x(t) = e 5t ( ) 6 / 21
x(t)? ẍ + 4ẋ + 3x = 0 ( ) 7 / 21
ẍ + 4ẋ + 3x = 0 x(t)? x(t) = e λt ẋ = λe λt ẍ = λλe λt = λ 2 e λt λ 2 e λt + 4λe λt + 3e λt = 0 λ 2 + 4λ + 3 = 0 λ = 3, 1 x(t) = e 3t x(t) = e t ( ) 7 / 21
ẍ + 4ẋ + 3x = 0 x(t)? x(t) = e λt ẋ = λe λt ẍ = λλe λt = λ 2 e λt λ 2 e λt + 4λe λt + 3e λt = 0 λ 2 + 4λ + 3 = 0 λ = 3, 1 x(t) = e 3t x(t) = e t ( ) 7 / 21
ẍ + 4ẋ + 3x = 0 x(t)? x(t) = C 1 e 3t x(t) = C 2 e t x(t) = C 1 e 3t + C 2 e t ẋ = 3C 1 e 3t C 2 e t, ẍ = 9C 1 e 3t + C 2 e t ẍ + 4ẋ + 3x = (9C 1 e 3t + C 2 e t ) + 4( 3C 1 e 3t C 2 e t ) + 3(C 1 e 3t + C 2 e t ) = C 1 (9 12 + 3)e 3t + C 2 (1 4 + 3)e t = 0 ( ) 8 / 21
x(t)? ẍ + 4ẋ + 3x = 0 x(t) = C 1 e 3t + C 2 e t (C 1, C 2 ) ( ) 9 / 21
ẍ + 4ẋ + 3x = 0 x(0) = 6, ẋ(0) = 2 x(t)? ( ) 10 / 21
ẍ + 4ẋ + 3x = 0 x(t)? x(0) = 6, ẋ(0) = 2 x(t) = C 1 e 3t + C 2 e t x(0) = C 1 + C 2 = 6 ẋ(0) = 3C 1 C 2 = 2 C 1 = 2, C 2 = 8 x(t) = 2e 3t + 8e t ( ) 10 / 21
ẍ + 9x = 0 ( ) 11 / 21
ẍ + 9x = 0 x(t) = e λt ẋ = λe λt ẍ = λλe λt = λ 2 e λt λ 2 e λt + 9e λt = 0 λ 2 + 9 = 0 λ = 3i, 3i x(t) = e 3it x(t) = e 3it x(t) = C 1 e 3it + C 2 e 3it ( ) 11 / 21
x(0) = 4, ẍ + 9x = 0 ẋ(0) = 6 ( ) 12 / 21
ẍ + 9x = 0 x(0) = 4, ẋ(0) = 6 x(0) = C 1 e 3i 0 + C 2 e 3i 0 = C 1 + C 2 = 4 ẋ(0) = (3i)C 1 e 3i 0 + ( 3i)C 2 e 3i 0 = 3i (C 1 C 2 ) = 6 C 1 = 2 + i, C 2 = 2 i x(t) = (2 + i)e 3it + (2 i)e 3it = (2 + i)(cos 3t + i sin 3t) + (2 i)(cos 3t i sin 3t) = 4 cos 3t 2 sin 3t ( ) 12 / 21
ẍ + 4ẋ + 13x = 0 ( ) 13 / 21
ẍ + 4ẋ + 13x = 0 x(t) = e λt ẋ = λe λt ẍ = λλe λt = λ 2 e λt λ 2 e λt + 4λe λt + 13e λt = 0 λ 2 + 4λ + 13 = 0 λ = 2 + 3i, 2 3i x(t) = e ( 2+3i)t x(t) = e ( 2 3i)t x(t) = C 1 e ( 2+3i)t + C 2 e ( 2 3i)t ( ) 13 / 21
x(0) = 4, ẍ + 4ẋ + 13x = 0 ẋ(0) = 2 ( ) 14 / 21
ẍ + 4ẋ + 13x = 0 x(0) = 4, ẋ(0) = 2 x(0) = C 1 + C 2 = 4 ẋ(0) = ( 2 + 3i)C 1 + ( 2 3i)C 2 = 2 C 1 = 2 i, C 2 = 2 + i x(t) = (2 i)e ( 2+3i)t + (2 + i)e ( 2 3i)t = e 2t {(2 i)(cos 3t + i sin 3t) + (2 + i)(cos 3t i sin 3t)} = e 2t (4 cos 3t + 2 sin 3t) ( ) 14 / 21
ẍ + 6ẋ + 9x = 0 ( ) 15 / 21
ẍ + 6ẋ + 9x = 0 x(t) = e λt λ 2 + 6λ + 9 = 0 λ = 3 ( ) x(t) = e 3t x(t) = Ce 3t ( ) 15 / 21
ẍ + 6ẋ + 9x = 0 C C(t) x(t) = C(t)e 3t ẋ = Ce 3t + C( 3)e 3t = ( C 3C)e 3t ẍ = ( C 3 C)e 3t + ( C 3C)( 3)e 3t = ( C 6 C + 9C)e 3t ( C 6 C + 9C) + 6( C 3C) + 9C = 0 C = 0 x(t) = te 3t C(t) = t ( ) 15 / 21
ẍ + 6ẋ + 9x = 0 x(t) = C 1 e 3t + C 2 te 3t (C 1, C 2 ) ( ) 15 / 21
1. (1) ẍ + 9ẋ + 20x = 0 x(0) = 1, ẋ(0) = 7 (2) ẍ + 2ẋ + 5x = 0 x(0) = 2, ẋ(0) = 2 ( π ) 2 (3) ẍ + x = 0 2 x(0) = 3, x(1) = 5 (4) ẋ + y = 0 ẏ x = 0 x(0) = 2, y(0) = 0 ( ) 16 / 21
2. (1) x(t) = 8e 7t (2) x(t) = 3e 4t, 5e 6t (3) x(t) = 5 sin 4t (4) x(t) = 2te t ( ) 17 / 21
ẍ + (5t)ẋ + 2e t x = 0 ẍ + 5ẋx + 3x = 0 ẍ + 5ẋx + 3x = 0 ẍ + (5t)ẋ + 2e t x = 0 ( ) 18 / 21
( ) ẍ + 5ẋ + 2x = 0 ( ) ẍ + 5ẋ + 2x = 4 sin 6t ( ) 19 / 21
ẋ + 5x = 0 ẍ + 4ẋ + 3x = 0... x 6ẍ + 11ẋ 6x = 0 ẋ + 5x = 0 ẋ + 5x = 2e 3t ẍ + 5ẋ + 2x = 4 sin 6t ( ) 20 / 21
Ce λt C 1 e λ1t + C 2 e λ 2t λ 1, λ 2 ( ) 21 / 21