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Application of Laplace Transform

s cos at
2
s +a 2
bt
a e sin at
2
(s−b) +a 2
bt
s−b e cos at
2
(s−b) +a 2
a sinh at
2
s −a 2
s cosh at
2
s −a 2
a e sinh at
bt
2
(s−b) −a 2
s−b e cosh at
bt
2
(s−b) −a 2
1 1 (sin at − at cos at)
2 2
2
(s +a ) 2a 3
s 1 (t sin at)
2 2
2
(s +a ) 2a
s 2 1 (sin at + at cos at)
2
2 2
(s +a ) 2a
s 3 cos at − at sin at
1
2 2
2
(s +a ) 2
2
s −a 2 t cos at
(s +a )
2
2 2
1 1 (at cosh at − sinh at)
2 2
2
(s −a ) 2a 3
s 1 (t sinh at)
2 2
2
(s −a ) a
s 2 1 (sinh at + at cosh at)
2
2 2
(s −a ) 2a
s 3 cosh at + at sinh at
1
2 2
2
(s −a ) 2
2
s +a 2 t cosh at
2
2 2
(s −a )
ab (a ̸= b ) a sin bt−b sin at
2
2
2
2
2
2
2
(s +a )(s +b ) a −b 2
s (a ̸= b ) cos bt−cos at
2
2
2
2
2
2
2
(s +a )(s +b ) a −b 2
2
2
s 2 (a ̸= b ) a sin at−b sin bt
2
2
2
2
2
(s +a )(s +b ) a −b 2
2
2
2
2
s 3 (a ̸= b ) a cos at−b cos bt
2
(s +a )(s +b ) a −b 2
2
2
2
2
ab (a ̸= b ) b sinh at−a sinh bt
2
2
2
(s −a )(s −b ) a −b 2
2
2
2
2
s (a ̸= b ) cosh at−cosh bt
2
2
2
(s −a )(s −b ) a −b 2
2
2
2
2
2
2
s 2 (a ̸= b ) a sinh at−b sinh bt
2
2
2
2
2
(s −a )(s −b ) a −b 2
2
2
s 3 (a ̸= b ) a cosh at−b cosh bt
2
2
2
2
2
2
2
(s −a )(s −b ) a −b 2
1
a 2 t − sin at
2
2
2
s (s +a ) a
a 2 1 sinh at − 1
2
2
2
s (s −a ) a
1 1
√ √
s πt
1 e −at
√ √
s+a πt √
1 1
√ √ erf( at)
s s+a a
1 e −bt −e −at
√ √ √
s+a+ s+b 2(b−a) πt 3
√
1 t
√ 2
s s π √
1 1 at
√ √ e erf at
(s−a) s s
( √ )
2
1 at 1 b t
√ e √ − be erfc(b t)
s−a+b πt
√ √ 1 bt at
s − a − s − b √ (e − e )
2t πt
102