What is inverse oflaplace 1/(s^2(s^2+3s+10)) ?

The answer to inverse oflaplace 1/(s^2(s^2+3s+10)) is

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inverse oflaplace 1/(s^2(s^2+3s+10))

inverse laplace

\frac{1}{s ^{2} ( s ^{2} + 3 s + 10 )}

- \frac{3}{100} H (t) + \frac{t}{10} + \frac{3}{100} e^{- \frac{3 t}{2}} cos (\frac{31 t}{2}) - \frac{11 e ^{- \frac{3 t}{2}} sin ( \frac{31 t}{2} )}{100 31}

Solution steps

L^{- 1} {\frac{1}{s ^{2} ( s ^{2} + 3 s + 10 )}}

Take the partial fraction of

\frac{1}{s ^{2} ( s ^{2} + 3 s + 10 )} : - \frac{3}{100 s} + \frac{1}{10 s ^{2}} + \frac{3 s - 1}{100 ( s ^{2} + 3 s + 10 )}

= L^{- 1} {- \frac{3}{100 s} + \frac{1}{10 s ^{2}} + \frac{3 s - 1}{100 ( s ^{2} + 3 s + 10 )}}

Expand

\frac{3 s - 1}{100 ( s ^{2} + 3 s + 10 )} : \frac{3}{100} \cdot \frac{s + \frac{3}{2}}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}} - \frac{11}{200} \cdot \frac{1}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}}

= L^{- 1} {- \frac{3}{100 s} + \frac{1}{10 s ^{2}} + \frac{3}{100} \cdot \frac{s + \frac{3}{2}}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}} - \frac{11}{200} \cdot \frac{1}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}}}

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{3}{100 s}} + L^{- 1} {\frac{1}{10 s ^{2}}} + \frac{3}{100} L^{- 1} {\frac{s + \frac{3}{2}}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}}} - \frac{11}{200} L^{- 1} {\frac{1}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}}}

L^{- 1} {\frac{3}{100 s}} : \frac{3}{100} H (t)

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

L^{- 1} {\frac{s + \frac{3}{2}}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}}} : e^{- \frac{3 t}{2}} cos (\frac{31 t}{2})

L^{- 1} {\frac{1}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}}} : e^{- \frac{3 t}{2}} \frac{2}{31} sin (\frac{31 t}{2})

= - \frac{3}{100} H (t) + \frac{t}{10} + \frac{3}{100} e^{- \frac{3 t}{2}} cos (\frac{31 t}{2}) - \frac{11}{200} e^{- \frac{3 t}{2}} \frac{2}{31} sin (\frac{31 t}{2})

Refine

- \frac{3}{100} H (t) + \frac{t}{10} + \frac{3}{100} e^{- \frac{3 t}{2}} cos (\frac{31 t}{2}) - \frac{11}{200} e^{- \frac{3 t}{2}} \frac{2}{31} sin (\frac{31 t}{2}) : - \frac{3}{100} H (t) + \frac{t}{10} + \frac{3}{100} e^{- \frac{3 t}{2}} cos (\frac{31 t}{2}) - \frac{11 e ^{- \frac{3 t}{2}} sin ( \frac{31 t}{2} )}{100 31}

= - \frac{3}{100} H (t) + \frac{t}{10} + \frac{3}{100} e^{- \frac{3 t}{2}} cos (\frac{31 t}{2}) - \frac{11 e ^{- \frac{3 t}{2}} sin ( \frac{31 t}{2} )}{100 31}

\frac{d}{d x} (\frac{4 e ^{2 x} - 2 e ^{x} + 1}{2 e ^{2 x}})

\int 3 x^{2} e^{x} d x

\frac{\partial}{\partial x} (- 2 y csc^{3} (x))

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

\int \frac{3}{2} x e^{- 6 x} d x

What is inverse oflaplace 1/(s^2(s^2+3s+10)) ?
The answer to inverse oflaplace 1/(s^2(s^2+3s+10)) is