to determine Laplace ftSolutionwe have Lft Fs here i m denot

to determine Laplace {f\'\'(t)}

Solution

we have L[f(t)]= F(s)

here i m denoting the sign of integration(including the limits) by \"?\"

now L[f\'(t)]= ?[e^-stf\'(t)dt] = [e^-stf(t)dt]₀ + s?[e^-stf(t)dt] ...........taking partial fraction

= sL[f(t)] + f(0)   

again L[f\"(t)] = sL[f\'(t)] - f\'(0)

= s[sL{f(t)} + f(0)] - f\'(0)

= s²L{f(t)} + sf(0) - f\'(0)

=s²F(s) + sf(0) - f\'(0)

hence the answer