Answer to Question #127086 in Differential Equations for jse

Question

1. Find the inverse Laplace transform ℒ-^1 { e^-5s / s^2 - 25 }

2.Find the Laplace transform of the given function.

f(t) = {1, 0 ≤ t < 6
{ 0, t ≥ 6

Enclose numerators and denominators in parentheses. For example, (a−b)/(1+n).


L {f(t)} = __________ , s>0.

Answer 1

$L(\sh(at))=\dfrac{a}{p^2-a^2}\newline \dfrac{e^{-5s}}{s^2-25}=\dfrac{1}{5}e^{-5s}\dfrac{5}{p^2-25}\newline L^{-1}(\dfrac{e^{-5s}}{s^2-25})=\dfrac{1}{5}\sh(5t)\eta(t-5)\newline Where \space \eta(t) \space is \space a\space Heaviside's function\newline \newline 2. f(t)=H_{0,6}(t)=\eta(t)-\eta(t-6)\newline L(f(t))=\dfrac{1}{s}-e^{-6s}\dfrac{1}{s}=\dfrac{1-e^{-6s}}{s}$