Lista de Resolução de EDO por Série de potência

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Universidade Federal de Sa˜o Paulo
Pro´-Reitoria de Graduac¸a˜o
Campus Diadema
Lista de Exerc´ıcios - Equac¸o˜es Diferenciais (Soluc¸a˜o por Se´ries de
Poteˆncia)
Nos problemas de 1 a 14, use se´ries de poteˆncias para resolver a equac¸a˜o diferencial.
1. y′ − y = 0
2. y′ = xy
3. y′ = x2y
4. (x− 3)y′ + 2y = 0
5. y′′ + xy′ + y = 0
6. y′′ = y
7. (x− 1)y′′ + y′ = 0
8. y′′ = xy
9. y′′ − xy′ − 2y = 0
10. y′′ + x2y = 0, y(0) = 1, y′(0) = 0
11. y′′ − xy′ − y = 0, y(0) = 1, y′(0) = 0
12. y′′ + x2y′ + xy = 0, y(0) = 0, y′(0) = 1
13. x2y′′ + xy′ + x2y = 0, y(0) = 1, y′(0) = 0
14. y′′ + xy′ + y = 0, y(0) = 0, y′(0) = 1
Respostas:
1. y(x) = C0
∞∑
n=0
xn
n!
2. y(x) = C0
∞∑
n=0
x2n
2n · n!
1
Universidade Federal de Sa˜o Paulo
Pro´-Reitoria de Graduac¸a˜o
Campus Diadema
3. y(x) = C0
∞∑
n=0
x3n
3n · n!
4. y(x) = C0
∞∑
n=0
(n + 1)
3n
xn
5. y(x) = C0
∞∑
n=0
(−1)n
2n · n!x
2n + C1
∞∑
n=0
(−2)n · n!
(2n + 1)!
x2n+1
6. y(x) = C0
∞∑
n=0
x2n
(2n)!
+ C1
∞∑
n=0
x2n+1
(2n + 1)!
7. y(x) = C0 + C1
∞∑
n=0
xn
n
8. y(x) = C0
∞∑
n=0
(3n− 2)(3n− 5) · · · 7 · 4 · 1
(3n)!
x3n +C1
∞∑
n=0
(3n− 1)(3n− 4) · · · 8 · 5 · 2
(3n + 1)!
x3n+1
9. y(x) = C0 + C0
∞∑
n=1
2n−1
(2n− 1)!x
2n + C1
∞∑
n=0
x2n+1
2n · n!
10. y(x) = 1 +
∞∑
n=1
(−1)n
4n(4n− 1)(4n− 4)(4n− 5) · · · 4 · 3x
4n
11. y(x) =
∞∑
n=0
x2n
2n · n!
12. y(x) = x +
∞∑
n=0
[
(−1)n · 22 · 52 · · · (3n− 1)2
(3n + 1)!
x3n+1
]
13. y(x) =
∞∑
n=0
(−1)nx2n
22n · (n!)2
14. y(x) =
∞∑
n=0
(−1)n · 2n · n!
(2n + 1)!
x2n+1
2