Nasab Teladan 1
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Pemecahan Persamaan differensial dengan metode deret kuasa dan kalkulus integral Siti mariam Siti maryam nur azizah Titin nurhayati Soal : y”+9y=0 solusi: Persamaan differensial denganMenggunakan Deret y =a0+a1x+a2x2+a3x3+a4x4+a5x5+ a6x6 + … y ' = a+2a x+3a x2+4a x3+5a x4+6a x5 +… 2
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y ' ' = 2a +6a x+12a x2+20a x3+30a x4+… 2 3 4 5 6 y ' '+9 y = 0 (2a2+6a3x+12a4x2+20a5x3+30a6x4+…)+9 (a0+a1x+a2x2+a3x3+a4x4+a5x5+ a6x6 + …) (2a2+6a3x+12a4x2+20a5x3+30a6x4+…)+(9 a0+9a1x+9a2x2+9a3x3+9a4x4+9a5x5+9 a6x6 -…) (2a2+9a0)+ (6a3+9a1)x+(12a4+9a2)x2+(20a5+9a3)x3+(30a5+9a5)x4+… (2a2+9 a0)= 0 → 2a2= -9 a0, 6a3+9a1=0
→ 6a -9a 3= 1,
12a4+9a2=0 → 12a4= -9a, 20a5+9a3=0
20a5=-9a3
9 − a0 2
a2=
9 − a1 a3= 6 a4=
−
a5=
9 9 9 81 a2 − ( − a0 ) 12 = 12 2 = 24 a0
−
9 9 9 81 a3 − ( − a1 a1 20 = 20 6 ) = 120
Jadi solusinya, y =a0+a1x+a2x2+a3x3+a4x4+a5x5+ a6x6 + … 9 9 81 81 a0 a1 a1 y =a0+a1x - 2 x2- 6 x3+ 24 a0x4+ 120 x5+… 9 81 9 81 + y= a0 (1- 2! x2 4! x4+…)+ a1(x - 3! x3+ 5! x5+…) y= a0 .cos3x+ a1.sin3x Persamaan differensial dengan Menggunakan kalkulus integral y ' '+9 y = 0 (D2+9)y =0 D2= -9 D= ± − 9 D1=3i D2=-3i y=C1.cos3x + C2sin3x
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y ' ' = 2a +6a x+12a x2+20a x3+30a x4+… 2 3 4 5 6 y ' '+9 y = 0 (2a2+6a3x+12a4x2+20a5x3+30a6x4+…)+9 (a0+a1x+a2x2+a3x3+a4x4+a5x5+ a6x6 + …) (2a2+6a3x+12a4x2+20a5x3+30a6x4+…)+(9 a0+9a1x+9a2x2+9a3x3+9a4x4+9a5x5+9 a6x6 -…) (2a2+9a0)+ (6a3+9a1)x+(12a4+9a2)x2+(20a5+9a3)x3+(30a5+9a5)x4+… (2a2+9 a0)= 0 → 2a2= -9 a0, 6a3+9a1=0
→ 6a -9a 3= 1,
12a4+9a2=0 → 12a4= -9a, 20a5+9a3=0
20a5=-9a3
9 − a0 2
a2=
9 − a1 a3= 6 a4=
−
a5=
9 9 9 81 a2 − ( − a0 ) 12 = 12 2 = 24 a0
−
9 9 9 81 a3 − ( − a1 a1 20 = 20 6 ) = 120
Jadi solusinya, y =a0+a1x+a2x2+a3x3+a4x4+a5x5+ a6x6 + … 9 9 81 81 a0 a1 a1 y =a0+a1x - 2 x2- 6 x3+ 24 a0x4+ 120 x5+… 9 81 9 81 + y= a0 (1- 2! x2 4! x4+…)+ a1(x - 3! x3+ 5! x5+…) y= a0 .cos3x+ a1.sin3x Persamaan differensial dengan Menggunakan kalkulus integral y ' '+9 y = 0 (D2+9)y =0 D2= -9 D= ± − 9 D1=3i D2=-3i y=C1.cos3x + C2sin3x
