Question

Laplace Transform of t ^{2}sin(2t)? ...?

Answer 1

By laplace transform definition of { t^n * f(t) } 

(-1)^n * ( d^n/ds^n ) F(s) 

t^2*sin(2t) <==== let's apply it 

(-1)^2 * ( d^2/ds^2 ) ( 2 / (s^2 + 2^2) ) 

1 * ( d^2/ds^2 ) ( 2 / (s^2 + 4) ) <===== let's find the first and second derivative : 

( (s^2 + 2^2) * 0 - 2 * (2s) / (s^2 + 4)^2 ) 

( - 4s) / (s^2 + 4)^2 ) <==== let's find the second derivative 

( (s^2 + 4)^2 * -4 - - 4s * 2 * (s^2 + 4) * 2s ) / (s^2 + 4)^4 ) 

( -4(s^2 + 4)^2 + 16s^2 * (s^2 + 4) ) / (s^2 + 4)^4 ) 

( -4(s^2 + 4) + 16s^2 ) / (s^2 + 4)^3 ) 

( -4s^2 - 16 + 16s^2 ) / (s^2 + 4)^3 ) 

( 12s^2 - 16 ) / (s^2 + 4)^3 )

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