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

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

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

inverse laplace

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

- \frac{3}{50} H (t) + \frac{t}{5} + \frac{3}{50} e^{- \frac{3 t}{2}} cos (\frac{31 t}{2}) - \frac{11 e ^{- \frac{3 t}{2}} sin ( \frac{31 t}{2} )}{50 31}

Solution steps

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

Take the partial fraction of

\frac{2}{s ^{2} ( s ^{2} + 3 s + 10 )} : - \frac{3}{50 s} + \frac{1}{5 s ^{2}} + \frac{3 s - 1}{50 ( s ^{2} + 3 s + 10 )}

= L^{- 1} {- \frac{3}{50 s} + \frac{1}{5 s ^{2}} + \frac{3 s - 1}{50 ( s ^{2} + 3 s + 10 )}}

Expand

\frac{3 s - 1}{50 ( s ^{2} + 3 s + 10 )} : \frac{3}{50} \cdot \frac{s + \frac{3}{2}}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}} - \frac{11}{100} \cdot \frac{1}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}}

= L^{- 1} {- \frac{3}{50 s} + \frac{1}{5 s ^{2}} + \frac{3}{50} \cdot \frac{s + \frac{3}{2}}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}} - \frac{11}{100} \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}{50 s}} + L^{- 1} {\frac{1}{5 s ^{2}}} + \frac{3}{50} L^{- 1} {\frac{s + \frac{3}{2}}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}}} - \frac{11}{100} L^{- 1} {\frac{1}{( s + \frac{3}{2} ) ^{2} + \frac{31}{4}}}

L^{- 1} {\frac{3}{50 s}} : \frac{3}{50} H (t)

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

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}{50} H (t) + \frac{t}{5} + \frac{3}{50} e^{- \frac{3 t}{2}} cos (\frac{31 t}{2}) - \frac{11}{100} e^{- \frac{3 t}{2}} \frac{2}{31} sin (\frac{31 t}{2})

Refine

- \frac{3}{50} H (t) + \frac{t}{5} + \frac{3}{50} e^{- \frac{3 t}{2}} cos (\frac{31 t}{2}) - \frac{11}{100} e^{- \frac{3 t}{2}} \frac{2}{31} sin (\frac{31 t}{2}) : - \frac{3}{50} H (t) + \frac{t}{5} + \frac{3}{50} e^{- \frac{3 t}{2}} cos (\frac{31 t}{2}) - \frac{11 e ^{- \frac{3 t}{2}} sin ( \frac{31 t}{2} )}{50 31}

= - \frac{3}{50} H (t) + \frac{t}{5} + \frac{3}{50} e^{- \frac{3 t}{2}} cos (\frac{31 t}{2}) - \frac{11 e ^{- \frac{3 t}{2}} sin ( \frac{31 t}{2} )}{50 31}

tan (f (x)) (x) = - 4 x + \frac{1}{2} x - 5, at x = 1

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

s (t) = - 9.8 t^{2} + 2 t + 1

y = (2 x + 3)^{5} + 10 x (2 x + 3)^{4}

\int e x^{2} + π d x

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