Tugas 5

MATA KULIAH:

“KALKULUS LANJUT”

DOSEN PENGAMPUH

Prof.Dr.P. Siagian, M.Pd

NAMA: HERI RISDIANTONIM : 8106172005

PENDIDIKAN MATEMATIKAPASCA SARJANA

UNIVERSITAS NEGERI MEDAN2010

1 . y ''+9( y )=0 , y (0 )= 3, y'(0 )= -5

Jawab

L (y’’+9(y)) = L (0) L (y’’)+ L (9(y)) = L (0)

s2(y(s)) – s y(0) – y’(0) + 9 y(s)=0

s2(y(s)) –3s –5 +9 y(s) = 0

s2(y(s)) +9 y(s) = 3s +5

y(s)(s2 + 9) = 3s + 5

y ( s )=3 s+5s2+9

y ( t )= L -1

( 3 s+5s2+9 )

y ( t )= L -1

( 3 ss2+9 ) +L -1

( 5s2+9 )

y(t) = 3 cos3t + 5 sin3t

2 . y ''−3 y '+4 y=0 , y( 0)= 1, y'(0 )= 5

Jawab

L (y’’- 3(y) + 4y) = L (0) L (y’’)- L (3y’)+ L 4y= L (0)

s2(y(s)) – s y(0) – y’(0) – 3(s y(s) - y (0))+ 4 y(s)=0

s2y(s) – s – 5 – 3 s y(s) +3 + 4y(s) = 0

s2y(s)– 3 s y(s) + 4y(s) = s + 2

y(s)(s2 – 3 s + 4)= s + 2

y ( s )= s+2s2−3 s+4

y ( t )= L -1

( s+2s2−3 s+4 )

y ( t )= L -1

( ss2−3 s+4 ) +L -1

( 2s2−3 s+4 )

y ( t )= L -1

( s−32

( s−32)2+7

4) +L -1

(3

2

( s−32)2+7

4) +L -1

( 2

( s−32)2+7

4 )y ( t )=

L -1

( s−32

( s−32)2+7

4) +

(7

2

( s−32)2+7

4)

y ( t )=e3

2 cos √ 74 t+ 7

2 e3

2 sin√ 74 t

3 .d2 qdt 2 +8 dq

dt+25 q=30 sin 2t ; q(0 )=q ' (0 )=0

Jawab

L ( d2 q

dt2 +8 dqdt

+25 q )= L (30 sin 2t )

L (y’’)- L (3y’)+ L 4y= L (0)

s2(y(s)) – s y(0) – y’(0) – 3(s y(s) - y (0))+ 4 y(s)=0

s2y(s) – s – 5 – 3 s y(s) +3 + 4y(s) = 0

s2y(s)– 3 s y(s) + 4y(s) = s + 2

y(s)(s2 – 3 s + 4)= s + 2

y ( s )= s+2s2−3 s+4

y ( t )= L -1

( s+2s2−3 s+4 )

y ( t )= L -1

( ss2−3 s+4 ) +L -1

( 2s2−3 s+4 )

y ( t )= L -1

( s−32

( s−32)2+7

4) +L -1

(3

2

( s−32)2+7

4) +L -1

( 2

( s−32)2+7

4 )y ( t )=

L -1

( s−32

( s−32)2+7

4) +

(7

2

( s−32)2+7

4)

y ( t )=e3

2 cos √ 74 t+ 7

2 e3

2 sin√ 74 t