Answer to Question #243674 in Calculus for JaytheCreator

Question #243674

Let β„’{𝑓(𝑑)} = 𝐹(𝑠). Show that 𝑓(𝑑) = βˆ’ 1 𝑑 β„’ βˆ’1 {𝐹 β€² (𝑠)} Thus, if we know how to invert 𝐹 β€² (𝑠) then we know how to invert 𝐹(𝑠). Use this information to find the Laplace inverse transform of (i) arctan ( π‘Ž 𝑠 ), (ii) ln ( 𝑠+π‘Ž π‘ βˆ’π‘Ž ). 


1
Expert's answer
2021-10-01T04:34:27-0400

F(s)=∫0∞eβˆ’tsf(t)dtF(s)=\int^{\infin}_0e^{-ts}f(t)dt


Fβ€²(s)=eβˆ’tsf(t)F'(s)=e^{-ts}f(t)

Lβˆ’1(F(s))=f(t)L^{-1}(F(s))=f(t)

Lβˆ’1(Fβ€²(s))=f(t)/(βˆ’t)L^{-1}(F'(s))=f(t)/(-t)

f(t)=βˆ’tLβˆ’1(Fβ€²(s))f(t)=-tL^{-1}(F'(s))


i)

Fβ€²(s)=a1+s2F'(s)=\frac{a}{1+s^2}

f(t)=βˆ’atsintf(t)=-atsint


ii)

Fβ€²(s)=1s+aβˆ’1sβˆ’aF'(s)=\frac{1}{s+a}-\frac{1}{s-a}

f(t)=βˆ’t(eβˆ’atβˆ’eat)f(t)=-t(e^{-at}-e^{at})


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