Find the Laplace transform for this function f(x)=(1+2ax)x^(-1/2) e^(ax)

Sonia Elliott 2022-10-21 Answered
Find the Laplace transform for this function
f ( x ) = ( 1 + 2 a x ) x 1 2 e a x
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Answers (2)

cdtortosadn
Answered 2022-10-22 Author has 19 answers
Remember the Laplace transform
L [ e a t t n ] = Γ ( n + 1 ) ( s a ) n + 1
Now, we have
L [ ( 1 + 2 a x ) x 1 / 2 e a x ]
= L [ e a x x 1 / 2 ] + 2 a L [ e a x x 1 / 2 ]
= Γ ( 1 2 + 1 ) ( s a ) 1 2 + 1 + 2 a Γ ( 1 2 + 1 ) ( s a ) 1 2 + 1
= Γ ( 1 2 ) s a + 2 a 1 2 Γ ( 1 2 ) ( s a ) 3 / 2
= π s a + a π ( s a ) 3 / 2
= s π ( s a ) 3 / 2
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propappeale00
Answered 2022-10-23 Author has 5 answers
f ( x ) = ( 1 + 2 a x ) x 1 2 e a x
f ( x ) = x 1 2 e a x + 2 a x 1 2 e a x
f ¯ ( s ) = L [ x 1 2 ; s a ] + 2 a L [ x 1 2 ; s a ]
But
L [ x 1 2 ; s ] = 0 e s x x 1 2 d x
let t = s x then d t = s . d x , t from 0
= 1 s 0 e t t 1 2 d t
= 1 s 0 e t t 1 2 1 d t = Γ ( 1 2 ) s = π s
So
L [ x 1 2 ; s a ] = π s a
And
L [ x 1 2 ; s ] = 0 e s x x 1 2 d x
let t = s x then d t = s . d x , t from 0
= 1 s 0 e t t 1 2 d t
= 1 s 3 2 0 e t t 3 2 1 d t = Γ ( 3 2 ) s 3 2 = 1 2 Γ ( 1 2 ) s 3 2 = 1 2 s π s
So
L [ x 1 2 ; s a ] = 1 2 ( s a ) π s a
Thus
f ¯ ( s ) = π s a + 2 a 1 2 ( s a ) π s a
f ¯ ( s ) = π s a + a ( s a ) π s a
f ¯ ( s ) = [ 1 + a ( s a ) ] π s a
f ¯ ( s ) = [ s ( s a ) ] π s a
f ¯ ( s ) = s π ( s a ) 3 2
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