Sistemas Lineares de Primeira Ordem

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1 Contenido 
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Constantes .......................................................................................................................... 10 
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𝜶 𝜷𝒊 .......................... 29 
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Unidad 1: 
Sistemas Lineales 
de primer orden 
Conceptos básicos 
y definiciones
Sistemas Lineales 
con coeficientes 
constantes
Exponencial de 
una Matriz
. 
 
 
𝑥1´ = 𝑎11𝑥1 + 𝑎12𝑥2 + ⋯ + 𝑎1𝑛𝑥𝑛 + 𝑓1
𝑥2´ = 𝑎21𝑥1 + 𝑎22𝑥2 + ⋯ + 𝑎2𝑛𝑥𝑛 + 𝑓2
⋮
𝑥𝑛
′ = 𝑎𝑛1𝑥1 + 𝑎𝑛2𝑥2 + ⋯ + 𝑎𝑛𝑛𝑥𝑛 + 𝑓𝑛
(´) 𝑡 𝑛 ≥ 1, 𝑎𝑖𝑗 = 𝑎𝑖𝑗(𝑡)
𝑓𝑖 = 𝑓𝑖(𝑡) 1 ≤ 𝑖, 𝑗 ≤ 𝑛
𝛼 ≤ 𝑡 ≤ 𝛽 𝑎𝑖𝑗(𝑡) 𝑓𝑖(𝑡) 𝑎𝑖𝑗(𝑡)
𝑎𝑖𝑗(𝑡) = 𝑎𝑖𝑗 = 𝑐𝑡𝑒.
. 
 
𝑓𝑖 = 𝑓𝑖(𝑡) ≡
0
𝑥1
′ = −𝑥2 + 𝑡 
𝑥2
′ = 9𝑥1
𝑋′ = 𝐴(𝑡)𝑋 + 𝐹(𝑡)
𝑋′ =
𝑑𝑋
𝑑𝑡
𝐴 𝐹
𝑓𝑛.
𝐴(𝑡) = (
𝑎11(𝑡) ⋯ 𝑎1𝑛(𝑡)
⋮ ⋱ ⋮
𝑎𝑛1(𝑡) ⋯ 𝑎𝑛𝑛(𝑡)
) 𝑋 = (
𝑥1
⋮
𝑥𝑛
) 𝑋′ = (
𝑥1′
⋮
𝑥𝑛′
) 𝐹(𝑡) = (
𝑓1(𝑡)
⋮
𝑓𝑛(𝑡)
)
. 
 
𝐴(𝑡)
 𝑥′ =
𝐴(𝑡)𝑥
 𝑥 = 𝑢 + 𝑖𝑣 
𝑋′ = 𝐴(𝑡)𝑋 + 𝐹(𝑡) 𝐹(𝑡) = 𝐺(𝑡) + 𝑖𝐻(𝑡) 𝐺(𝑡) 𝐻(𝑡)
𝑢′ = 𝐴(𝑡)𝑢 + 𝐺(𝑡) 
𝑣′ = 𝐴(𝑡)𝑣 + 𝐻(𝑡)
 𝑢 𝑣 𝑥 =
𝑢 + 𝑖𝑣 𝑋′ = 𝐴(𝑡)𝑋 + 𝐹(𝑡) 𝐹(𝑡) = 𝐺(𝑡) + 𝑖𝐻(𝑡)
𝑢, 𝑣, 𝑔, ℎ
(𝑢 + 𝑖𝑣)′ = 𝐴(𝑡)(𝑢 + 𝑖𝑣) + 𝐺(𝑡) + 𝑖𝐻(𝑡)
𝑢′ = 𝐴(𝑡)𝑢 + 𝐺(𝑡) 
𝑣′ = 𝐴(𝑡)𝑣 + 𝐻(𝑡)
. 
 
𝑔 ≡ ℎ ≡ 0
𝑖 𝑣′ = 𝐴(𝑡)𝑣 + 𝐻(𝑡) 𝑢′ =
𝐴(𝑡)𝑢 + 𝐺(𝑡)
𝑋′ = 𝐴(𝑡)𝑋 + 𝐹(𝑡)
𝐿(𝑋) = 𝑋′ − 𝐴(𝑡)𝐼𝑋 = 𝐹(𝑡)
𝐹(𝑡) 𝑋
𝑛 𝛼 < 𝑡 < 𝛽 𝐿
𝐿(𝑢 + 𝑣) = 𝐿(𝑢) + 𝐿(𝑣) 
𝐿(𝛾𝑢) = 𝛾𝐿(𝑢)
𝑢, 𝑣 𝐿 𝑋
𝐿
𝑥1(𝑡), … , 𝑥𝑛(𝑡)
𝑐1, … , 𝑐𝑛
𝑡
𝑐1𝑥1(𝑡) + ⋯ + 𝑐𝑛𝑥𝑛(𝑡) ≡ 0.
. 
 
𝑊(𝑡) = |
𝑥11(𝑡) ⋯ 𝑥1𝑛(𝑡)
⋮ ⋱ ⋮
𝑥𝑛1(𝑡) ⋯ 𝑥𝑛𝑛(𝑡)
|
𝑥1(𝑡), … , 𝑥𝑛(𝑡)
𝑊(𝑡) ≡ 0.
𝑛
𝑋′ = 𝐴(𝑡)𝑋 + 𝐹(𝑡)
𝑓𝑖
𝐼 ⊂ ℝ 𝑖 ∈ ℕ ∈ 𝐼 𝑥 ∈ ℝ
𝑥 𝐼
. 
 
𝑥1(𝑡), … , 𝑥𝑛(𝑡) 𝑛 𝑋
′ = 𝐴(𝑡)𝑋
𝑥(𝑡) = 𝑐1𝑥1(𝑡) + ⋯ + 𝑐𝑛𝑥𝑛(𝑡),
𝑐1, … , 𝑐𝑛
𝑛
𝑋(𝑡)
𝑋(𝑡)
𝑥(𝑡) = 𝑐𝑋(𝑡) 𝑐
. 
 
𝑐 = (
𝑐1
⋮
𝑐𝑛
).
𝑥′ = 𝑦 
𝑦′ = 0.
𝑦 = 𝑐1
𝑐1
𝑥′ = 𝑐1,
𝑥 = 𝑐1𝑡 + 𝑐2.
𝑥 = 𝑐1𝑡 + 𝑐2, 
𝑦 = 𝑐1.
. 
 
𝑐1 = 1 𝑐2 = 0
𝑥1 = 𝑡, 
𝑦1 = 1.
𝑐1 = 0 𝑐2 = 1
𝑥2 = 1, 
𝑦2 = 0.
𝑊(𝑡) = |
𝑥1 𝑥2
𝑦1 𝑦2
| = |
𝑡 1
1 0
| = −1 ≠ 0.
𝑋(𝑡) = (
𝑥1 𝑥2
𝑦1 𝑦2
) = (
𝑡 1
1 0
).
𝑥1(𝑡) = 𝑐1𝑒
𝑡 
𝑥2(𝑡) = 𝑐1𝑒
𝑡 + 𝑐2𝑒
2𝑡
𝑐1, 𝑐2 ∈ ℝ 𝑘
𝑥1
′ = 𝑥1 
𝑥2
′ = −𝑥1 + 2𝑥2
. 
 
𝑋 ′ 𝐴𝑋 𝐹
𝐴 = (
1 0
−1 2
)
𝑋 = (
𝑐1𝑒
𝑡
𝑐1𝑒
𝑡 + 𝑐2𝑒
2𝑡)
𝐹 = (
0
0
)
𝑋´ =
𝑑𝑋
𝑑𝑡
𝐴𝑋
𝑋′ = (
𝑐1𝑒
𝑡
𝑐1𝑒
𝑡 + 2𝑐2𝑒
2𝑡) 
𝐴𝑋 = (
𝑐1𝑒
𝑡
𝑐1𝑒
𝑡 + 2𝑐2𝑒
2𝑡) 
. 
 
𝑋′ = 𝐴𝑋.
𝑥𝑔
 𝑥𝑝 𝑥0
𝑢′ = 𝐴(𝑡)𝑢
𝑥𝑔 = 𝑥𝑝 + 𝑥0
𝑋′ = 𝐴(𝑡)𝑋 + 𝐹(𝑡)
𝐹(𝑡) = 0
𝑐1, … , 𝑐𝑛
𝑐1(𝑡), … , 𝑐𝑛(𝑡)
𝑥(𝑡) = 𝑐1(𝑡)𝑥1 + ⋯ + 𝑐𝑛(𝑡)𝑥𝑛,
𝑐1(𝑡), … , 𝑐𝑛(𝑡)
. 
 
𝑥(𝑡) = 𝑐(𝑡)𝑋(𝑡),
𝑥′ = 𝐴(𝑡)𝑥 + 𝐹(𝑡)
𝑋′𝑐 + 𝑋𝑐′ = 𝑐𝐴𝑋 + 𝑓
𝑋′ = 𝐴𝑋
𝑋𝑐′(𝑡) = 𝐹(𝑡)
𝑐′(𝑡) = 𝑋−1(𝑡)𝑓(𝑡)
𝑐(𝑡) = 𝑐0 + ∫ 𝑋
−1(𝑠)𝑓(𝑠)𝑑𝑠
𝑡
𝑡0
𝑐0
𝑥(𝑡) = 𝑐(𝑡)𝑋(𝑡) = 𝑐0𝑋(𝑡) + 𝑋(𝑡) ∫ 𝑋
−1(𝑠)𝑓(𝑠)𝑑𝑠
𝑡
𝑡0
.
. 
 
(
𝑥
𝑦) = 𝑐1 (
sin 𝑡
cos 𝑡
) + 𝑐2 (
cos 𝑡
− sin 𝑡
), 
𝑥′ = 𝑦 
𝑦′ = −𝑥
𝑥′ = 𝑦 
𝑦′ = −𝑥 +
1
sin 𝑡
0 < 𝑡 < 𝜋.
𝑐1 𝑐2 𝑐1′ 𝑐2′
𝑐1
′ (
sin 𝑡
cos 𝑡
) + 𝑐2
′ (
cos 𝑡
− sin 𝑡
) = (
0
1/ sin 𝑡
),
𝑐1′ sin 𝑡 + 𝑐2
′ cos 𝑡 = 0 
𝑐1′ cos 𝑡 + 𝑐2
′ sin 𝑡 = 1/ sin 𝑡
. 
 
𝑐1
′ =
cos 𝑡
sin 𝑡
𝑐2
′ = −1
𝑐1 = ln sin 𝑡 + 𝑐11 
𝑐2 = −𝑡 + 𝑐21
(
𝑥
𝑦) = (ln sin 𝑡 + 𝑐11) (
sin 𝑡
cos 𝑡
) + (−𝑡 + 𝑐21) (
cos 𝑡
− sin 𝑡
).
𝑥′ = 𝑎𝑥 + 𝑏𝑦 
𝑦′ = 𝑐𝑥 + 𝑑𝑦
𝑋′ = 𝐴𝑋
𝐴 = (
𝑎 𝑏
𝑐 𝑑
)
𝑎, 𝑏, 𝑐, 𝑑 
𝑋(𝑡) = (
𝑥(𝑡)
𝑦(𝑡)
)
𝑥𝑦
(𝑥, 𝑦)
𝑡
. 
 
𝑥𝑦 𝑥𝑦
(𝑥0, 𝑦0) 𝑋
′ = 𝐴𝑋
(
𝑥′
𝑦′
) = 𝐴 (
𝑦0
𝑥0
) = (
0
0
).
𝑋′ = 𝐴𝑋
. 
 
𝐴 = (
−1 −1
1 −1
).
𝑥′ = −𝑥 − 𝑦, 
𝑦′ = 𝑥 − 𝑦.
𝑑𝑦
𝑑𝑥
𝑥 𝑦 𝑥′ 𝑦′
 𝑥′ > 0 𝑦′ > 0
𝑑𝑦
𝑑𝑥
> 0
 𝑥′ > 0 𝑦′ < 0,
𝑑𝑦
𝑑𝑥
< 0
 𝑥′ < 0 𝑦′ < 0,
𝑑𝑦
𝑑𝑥
> 0
 𝑥′ < 0 𝑦′ < 0,
𝑑𝑦
𝑑𝑥
> 0
. 
 
 𝐴 𝑛 × 𝑛 𝜆
𝐴 𝑢 = (𝑥1, 𝑥2) ≠ 0
𝐴𝑢 = 𝜆𝑢.
𝑢 𝐴 𝜆.
. 
 
𝑛 × 𝑛
𝑃𝐴(𝜆) = det(𝐴 − 𝜆𝐼) = 0,
𝜆 𝐴 𝐼
𝐴 = (
𝑎 𝑏
𝑐 𝑑
),
det(𝐴 − 𝜆𝐼) = det ((
𝑎 𝑏
𝑐 𝑑
) − 𝜆 (
1 0
0 1
)) 
𝜆2 − (𝑎 + 𝑑)𝜆 + (𝑎𝑑 − 𝑏𝑐) = 0 
𝐴.
. 
 
𝑋′ = (
2 1
2 3
) 𝑋 
𝜆2 − 5𝜆 + 4 = 0 
𝜆1 = 4 
𝜆2 = 1
𝐴𝑢 = 𝜆𝑢,
(
2 1
2 3
) (
𝑥1
𝑥2
) = 4 (
𝑥1
𝑥2
)
2𝑥1 + 𝑥2 = 4𝑥1 
. 
 
2𝑥1 + 3𝑥2 = 4𝑥2
−2𝑥1 + 𝑥2 = 0 
2𝑥1 − 𝑥2 = 0
2𝑥1 − 𝑥2 = 0
2𝑥1 = 𝑥2
𝑢 = (1,2).
𝜆 = 1
𝑥1 = 𝑥2
𝜆 = 1, 𝑢 = (1,1).
𝜆 𝑢
𝑋′ = 𝐴𝑋
. 
 
𝑋(𝑡) = 𝑒𝜆𝑡𝑢 
𝐴 𝜆1 𝜆2 𝑢1 𝑢2
𝑋´ = 𝐴𝑋
𝑋1 = exp(𝜆1𝑡) 𝑢1 
𝑋2 = exp (𝜆2𝑡)𝑢2
𝑋(𝑡) = 𝑐1 exp(𝜆1𝑡)𝑢1 + 𝑐2 exp(𝜆2𝑡)𝑢2 
𝑐1, 𝑐2 ∈ ℝ
 𝜆
𝜆.
 𝜆
𝜆
. 
 
𝜆 𝜆
𝜆 𝜆
𝑋′ = (
2 1
2 3
) 𝑋 
λ λ
𝑋(𝑡) = 𝑐1 exp(4𝑡) (
1
2
) + 𝑐2 exp(𝑡) (
1
1
)
𝑐1, 𝑐2 ∈ ℝ. 
. 
 
𝜆1 > 0 > 𝜆2 
𝜆2
𝜆1
𝑋′ = (
−2 0
3 4
) 𝑋 
𝜆1 = −2
𝑢 = (2,1) 𝜆2 = 4 𝑢 = (1, 0).
. 
 
𝑋(𝑡) = 𝑐1 exp(−2𝑡) (
2
1
) + 𝑐2 exp(4𝑡) (
1
0
) 
𝑐1, 𝑐2 ∈ ℝ. 
. 
 
𝐴 𝜆1, … , 𝜆𝑛 𝑢1, … , 𝑢𝑛
𝑢1, … , 𝑢𝑛
𝐴 𝜆1, … , 𝜆𝑛 𝑢1, … , 𝑢𝑛
exp(𝜆1𝑡) 𝑢1, … , exp(𝜆𝑛𝑡) 𝑢𝑛
𝑋’ = 𝐴𝑋
𝐴 ℝ
𝛼 𝛽𝑖 𝑏𝑖
. 
 
𝑋 𝐴𝑋
𝑥1 = exp(𝛼𝑡) cos(𝛽𝑡) 𝑎 − exp(𝛼𝑡) sin(𝛽𝑡) 𝑏 
𝑥2 = exp(𝛼𝑡) sin(𝛽𝑡) 𝑎 + exp(𝛼𝑡) cos(𝛽𝑡) 𝑏. 
𝜶 𝜷𝒊
𝛼
𝛼
𝛼
𝑋’ = (
3 −1
5 −1
) 𝑋 
𝜆2 − 2𝜆 + 2 = 0
. 
 
𝜆 = 1 ± 𝑖. 𝜆1 = 1 + 𝑖
𝜆2 = 1 − 𝑖.
𝐴𝑢 = 𝜆 𝑢 𝜆1
(
3 −1
5 −1
) (
𝑥1
𝑥2
) = (1 + 𝑖) (
𝑥1
𝑥2
) 
𝑥2 = (3 − 𝑖)𝑥1 𝑢 = (1,3 − 𝑖).
𝛼 = 1 𝛽 = 1
𝑢 = (
1
3 − 𝑖
) = (
1
3
) + 𝑖 (
0
−1
),
𝑎 = (
1
3
) 𝑏 = (
0
−1
).
𝑋(𝑡) = 𝑐1𝑒
𝑡 (cos 𝑡 (
1
3
) − sin 𝑡 (
0
−1
) ) 
+𝑐2𝑒
𝑡 (sin 𝑡 (
1
3
) − cos 𝑡 (
0
−1
) ).
 𝑐1, 𝑐2 ∈ ℝ
. 
 
𝐴 𝑋′ = 𝐴𝑋 𝜆 𝑢1
𝑋1(𝑡) = exp(𝜆𝑡) 𝑢1
𝑋2(𝑡) = exp(𝜆𝑡) (𝑢1𝑡 +
𝑢2) 𝑋1(𝑡) 𝑢2 𝐴𝑢2 − 𝜆𝑢2 = 𝑢1.
𝑋(𝑡) = 𝑐1 exp(𝜆𝑡) 𝑢1 + 𝑐2 exp(𝜆𝑡)(𝑢1𝑡 + 𝑢2) 
. 
 
𝑐1, 𝑐2 ∈ ℝ.
𝜆
𝜆
𝑋′ = (
−4 −2
2 0
) 𝑋 
𝜆2 + 4𝜆 + 4 = 0
𝜆 = −2.
𝑢 = (𝑥1, −𝑥1),
𝜆 = 2 𝑢 = (1, −1).
𝑥1(𝑡) = 𝑐1𝑒
−2𝑡𝑢1
. 
 
𝑢2 = (𝑧, 𝑤)
𝐴𝑢2 − 𝜆𝑢2 = 𝑢1 
(
−2 −2
2 2
) (
𝑧
𝑤
) = (
1
−1
)
𝑧 = −1/2 𝑤 = 0 𝑢 = (−
1
2
, 0).
𝑥2(𝑡) = 𝑐2𝑒
−2𝑡 [(
1
−1
) 𝑡 + (
−1/2
0
)]
𝑋(𝑡)= 𝑒2𝑡 {𝑐1 (
1
−1
) + 𝑐2 [(
1
−1
) 𝑡 + (
1/2
0
)]}.
. 
 
𝑝(𝑥) = 𝑎0 + 𝑎1𝑥 + ⋯ + 𝑎𝑛𝑥
𝑛
𝑝(𝐴) = 𝑎0𝐼 + 𝑎1𝐴 + ⋯ + 𝑎𝑛𝐴
𝑛
𝐴 𝑛 × 𝑛 𝐼 𝑛 × 𝑛
𝐴(𝑡) = (𝑎𝑖𝑗(𝑡))
𝑖,𝑗=1,…,𝑛
 
𝐴′(𝑡) = (𝑎´𝑖𝑗(𝑡))
𝑖,𝑗=1,…,𝑛
.
. 
 
𝑛 × 𝑛 𝐴 exp(𝐴) 𝑛 × 𝑛
𝑒𝐴 = 𝐼 + 𝐴 +
𝐴2
2!
+ ⋯ +
𝐴𝑛
𝑛!
 𝐴𝐵 = 𝐵𝐴 𝑒𝐴+𝐵 = 𝑒𝐴𝑒𝐵.
 
𝑑
𝑑𝑡
𝑒𝑡𝐴 = 𝐴𝑒𝑡𝐴
𝑋(𝑡) = 𝑒𝑡𝐴
𝑑𝑋
𝑑𝑡
= 𝐴𝑋,
𝑋(0) = 𝐼. 𝑒𝑡𝐴
. 
 
𝑒𝐴𝑡 𝑥𝑘(𝑡) 𝑘 = 1, … , 𝑛
𝑋′ = 𝐴𝑋 𝑥𝑘(0) 𝑘
𝑘
𝑒𝐴𝑡
𝑋′ = 𝐴𝑋 𝑋(0) = (2,3) 𝑒𝑡𝐴 𝐴 = (
2 1
0 2
).
𝐴
𝐴 = (
2 1
0 2
) = (
2 0
0 2
) + (
0 1
0 0
) = 𝐵 + 𝐶,
exp(𝐴𝑡) = exp(𝐵𝑡) exp(𝐶𝑡),
exp(𝐴𝑡) = 𝑒2𝑡 (
1 𝑡
0 1
)
𝑋 = 𝑒𝐴𝑡𝑋(0)
𝑋 = 𝑒2𝑡 (
1 𝑡
0 1
) (
2
3
)
. 
 
𝑥1 = (3𝑡 + 2)𝑒
2𝑡 
𝑥2 = 3𝑒
2𝑡 .
 
. 
 
http://www.ehu.eus/izaballa/Ecu_Dif/Apuntes/lec8.pdf
http://www.ehu.eus/izaballa/Ecu_Dif/Apuntes/lec9.pdf
http://deymerg.files.wordpress.com/2011/07/capitulo-8.pdf
http://www2.caminos.upm.es/departamentos/matematicas/Fdistancia/PIE/Analisis%20matematico/Temas/C11_Sistemas.pdf
http://www2.caminos.upm.es/departamentos/matematicas/Fdistancia/PIE/Analisis%20matematico/Temas/C11_Sistemas.pdf
http://personal.us.es/niejimjim/tema03.pdf
. 
 
http://www.scottsarra.org/applets/dirField2/dirField2.html
http://math.rice.edu/~dfield/dfpp.html