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f-is-a-2-derivable-function-and-L-laplace-transfom-determine-L-f-a-L-f-

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Question Number 94931 by mathmax by abdo last updated on 22/May/20
f is a 2(×) derivable function and L laplace transfom determine L(f^′ ) a L(f^(′′) )
fisa2(×)derivablefunctionandLlaplacetransfomdetermineL(f′)aL(f″)
Answered by mathmax by abdo last updated on 22/May/20
L(f^′ (t))(p) =∫_0 ^∞ f^′ (t)e^(−pt) dt and by parts =[f(t)e^(−pt) ]_0 ^∞ −∫_0 ^∞ f(t)(−p)e^(−pt) dt =−f(0) +p ∫_0 ^∞ f(t)e^(−pt) dt =pL(f)−f(0^+ ) ⇒L(f^′ ) =pL(f)−f(0^+ ) also L(f^(′′) ) =L((f^′ )^′ ) =p L(f^′ )−f^′ (0^+ ) =p(pL(f)−f(0^+ ))−f^′ (0^+ ) =p^2 L(f)−pf(0^+ )−f^′ (0^+ )
L(f′(t))(p)=∫0∞f′(t)e−ptdtandbyparts=[f(t)e−pt]0∞−∫0∞f(t)(−p)e−ptdt=−f(0)+p∫0∞f(t)e−ptdt=pL(f)−f(0+)⇒L(f′)=pL(f)−f(0+)alsoL(f″)=L((f′)′)=pL(f′)−f′(0+)=p(pL(f)−f(0+))−f′(0+)=p2L(f)−pf(0+)−f′(0+)

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