What is inverse oflaplace 5/(s*(2*s+1)) ?

The answer to inverse oflaplace 5/(s*(2*s+1)) is 5\H(t)-5e^{-t/2}

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inverse oflaplace 5/(s(2s+1))

inverse laplace

\frac{5}{s \cdot ( 2 \cdot s + 1 )}

5 H (t) - 5 e^{- \frac{t}{2}}

Solution steps

L^{- 1} {\frac{5}{s ( 2 s + 1 )}}

Take the partial fraction of

\frac{5}{s ( 2 s + 1 )} : \frac{5}{s} - \frac{10}{2 s + 1}

= L^{- 1} {\frac{5}{s} - \frac{10}{2 s + 1}}

Use the linearity property of Inverse Laplace Transform:

For functions

f (s), g (s)

and constants

a, b : L^{- 1} {a \cdot f (s) + b \cdot g (s)} = a \cdot L^{- 1} {f (s)} + b \cdot L^{- 1} {g (s)}

= L^{- 1} {\frac{5}{s}} - L^{- 1} {\frac{10}{2 s + 1}}

L^{- 1} {\frac{5}{s}} : 5 H (t)

L^{- 1} {\frac{10}{2 s + 1}} : 5 e^{- \frac{t}{2}}

= 5 H (t) - 5 e^{- \frac{t}{2}}

3 y^{^{'}} - 6 y = 12, y (0) = 4

\int_{- 3}^{0} x^{2} - 4 x + 7 d x

\int x (- 4 - 7 x^{4}) d x

6 t \frac{d y}{d t} + y = t^{4}

\frac{d y}{d x} = \frac{1}{2} x + 3

What is inverse oflaplace 5/(s*(2*s+1)) ?
The answer to inverse oflaplace 5/(s*(2*s+1)) is 5\H(t)-5e^{-t/2}