Find inverse Laplace of an exponential function exp{-x sqrt((s+h)/k)}

George Morales

George Morales

Answered question

2022-10-28

Inverse Laplace of an exponential function exp { x ( s + h ) / k }

Answer & Explanation

elulamami

elulamami

Beginner2022-10-29Added 22 answers

Observe that if L { f ( t ) } = F ( s ), we have L { e h t f ( t ) } = F ( s + h ) and L { f ( k t ) } = 1 k F ( s k ) for k>0 and then
L { e h t f ( k t ) } = 1 k F ( s + h k )
so for f ( t ) = a 2 π t 3 exp ( a 2 / 4 t ) with Laplcae transform F ( s ) = exp ( a s ) we have
L { e h t f ( k t ) } = 1 k exp ( a s + h k )
and then for a=x we have
L 1 { exp ( x s + h k ) } = e h t k f ( k t ) = e h t k x 2 π ( k t ) 3 exp ( x 2 4 ( k t ) ) = x 2 π k t 3 exp ( h t x 2 4 k t )

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