Nasab Jadi
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Soal: Tentukan penyelesaian PD di bawah ini dengan menggunakan penyelesaian PD dan deret pangkat ! y’’-y’-6y = 0 Jawab: Penyelesaian PD Misal: y =
y’=b
y’’=
Maka: y’’ – y’-6y = 0 -b
-6
=0
(b – 3)(b + 2) = 0 b1 = 3 atau b2= -2 Jadi penyelesaian PDnya adalah y =
dan y =
Solusi deret pangkat Misalkan solusinya:
y’ =
+5
y’’ =
substitusikan ke persamaannya: ó
)–
+5
) – 6(
ó (2
+(
•
2a2 – a1 = 6a0
2a2 = 6a0 + a1 2a2 = 6a0 + a1
a2 = 3a0 + ½ a1 •
6a3 = 2a2 + 6a1 6a3 = 6a0 + a1 +6a1 6a3 = 6a0 +7a1 a3 = a0 +7/6 a1
•
12a4 = 3a3 + 6a2 12a4 = (3a0 +7/6 a1)+18a0 + 3a1 12a4 = 21a0 + 13/2 a1 a4 =
a4 = a0 + •
a0 +
a1
a1
20a5 = 4a4 + 6a3
= (
= 7a0 +
a0 +
a1)+ 6 (a0 +
a1 + 6a0 + 7a1
a1)
20a5 =13a0 + a5 =
a0 +
a5 =
a0 +
a5 =
a0 +
a1 a1
a1 a1
y = a0+ a1x + (2a0 + a1)x2 + (a0 + a1) x3 + … = a0 (1 + 3x2+ x3) + a1 (x +
x2 +
x3) + …
y’=b
y’’=
Maka: y’’ – y’-6y = 0 -b
-6
=0
(b – 3)(b + 2) = 0 b1 = 3 atau b2= -2 Jadi penyelesaian PDnya adalah y =
dan y =
Solusi deret pangkat Misalkan solusinya:
y’ =
+5
y’’ =
substitusikan ke persamaannya: ó
)–
+5
) – 6(
ó (2
+(
•
2a2 – a1 = 6a0
2a2 = 6a0 + a1 2a2 = 6a0 + a1
a2 = 3a0 + ½ a1 •
6a3 = 2a2 + 6a1 6a3 = 6a0 + a1 +6a1 6a3 = 6a0 +7a1 a3 = a0 +7/6 a1
•
12a4 = 3a3 + 6a2 12a4 = (3a0 +7/6 a1)+18a0 + 3a1 12a4 = 21a0 + 13/2 a1 a4 =
a4 = a0 + •
a0 +
a1
a1
20a5 = 4a4 + 6a3
= (
= 7a0 +
a0 +
a1)+ 6 (a0 +
a1 + 6a0 + 7a1
a1)
20a5 =13a0 + a5 =
a0 +
a5 =
a0 +
a5 =
a0 +
a1 a1
a1 a1
y = a0+ a1x + (2a0 + a1)x2 + (a0 + a1) x3 + … = a0 (1 + 3x2+ x3) + a1 (x +
x2 +
x3) + …
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