What is inverse oflaplace ((1-s^{-6})^2)/((1-s^{-1))^2} ?

The answer to inverse oflaplace ((1-s^{-6})^2)/((1-s^{-1))^2} is

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inverse oflaplace ((1-s^{-6})^2)/((1-s^{-1))^2}

inverse laplace

\frac{( 1 - s ^{- 6} ) ^{2}}{( 1 - s ^{- 1} ) ^{2}}

δ (t) + 2 H (t) + 3 t + 2 t^{2} + \frac{5 t ^{3}}{6} + \frac{t ^{4}}{4} + \frac{t ^{5}}{24} + \frac{t ^{6}}{180} + \frac{t ^{7}}{1680} + \frac{t ^{8}}{20160} + \frac{t ^{9}}{362880}

Solution steps

L^{- 1} {\frac{( 1 - s ^{- 6} ) ^{2}}{( 1 - s ^{- 1} ) ^{2}}}

Expand

\frac{( 1 - s ^{- 6} ) ^{2}}{( 1 - s ^{- 1} ) ^{2}} : 1 + \frac{2}{s} + \frac{3}{s ^{2}} + \frac{4}{s ^{3}} + \frac{5}{s ^{4}} + \frac{6}{s ^{5}} + \frac{5}{s ^{6}} + \frac{4}{s ^{7}} + \frac{3}{s ^{8}} + \frac{2}{s ^{9}} + \frac{1}{s ^{10}}

= L^{- 1} {1 + \frac{2}{s} + \frac{3}{s ^{2}} + \frac{4}{s ^{3}} + \frac{5}{s ^{4}} + \frac{6}{s ^{5}} + \frac{5}{s ^{6}} + \frac{4}{s ^{7}} + \frac{3}{s ^{8}} + \frac{2}{s ^{9}} + \frac{1}{s ^{10}}}

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} {1} + L^{- 1} {\frac{2}{s}} + L^{- 1} {\frac{3}{s ^{2}}} + L^{- 1} {\frac{4}{s ^{3}}} + L^{- 1} {\frac{5}{s ^{4}}} + L^{- 1} {\frac{6}{s ^{5}}} + L^{- 1} {\frac{5}{s ^{6}}} + L^{- 1} {\frac{4}{s ^{7}}} + L^{- 1} {\frac{3}{s ^{8}}} + L^{- 1} {\frac{2}{s ^{9}}} + L^{- 1} {\frac{1}{s ^{10}}}

L^{- 1} {1} : δ (t)

L^{- 1} {\frac{2}{s}} : 2 H (t)

L^{- 1} {\frac{3}{s ^{2}}} : 3 t

L^{- 1} {\frac{4}{s ^{3}}} : 2 t^{2}

L^{- 1} {\frac{5}{s ^{4}}} : \frac{5 t ^{3}}{6}

L^{- 1} {\frac{6}{s ^{5}}} : \frac{t ^{4}}{4}

L^{- 1} {\frac{5}{s ^{6}}} : \frac{t ^{5}}{24}

L^{- 1} {\frac{4}{s ^{7}}} : \frac{t ^{6}}{180}

L^{- 1} {\frac{3}{s ^{8}}} : \frac{t ^{7}}{1680}

L^{- 1} {\frac{2}{s ^{9}}} : \frac{t ^{8}}{20160}

L^{- 1} {\frac{1}{s ^{10}}} : \frac{t ^{9}}{362880}

= δ (t) + 2 H (t) + 3 t + 2 t^{2} + \frac{5 t ^{3}}{6} + \frac{t ^{4}}{4} + \frac{t ^{5}}{24} + \frac{t ^{6}}{180} + \frac{t ^{7}}{1680} + \frac{t ^{8}}{20160} + \frac{t ^{9}}{362880}

\frac{\partial}{\partial y} (x z ln (x + y + z))

\frac{d ^{2}}{d x ^{2}} (5 cos (2 x + 10))

q \to 18 lim (0.9 q^{2} - 6 q + 80)

sec (θ) (θ - tan (θ))

x^{2} + 8

What is inverse oflaplace ((1-s^{-6})^2)/((1-s^{-1))^2} ?
The answer to inverse oflaplace ((1-s^{-6})^2)/((1-s^{-1))^2} is