What is inverse oflaplace ((2))/(s^2+5s+3) ?

The answer to inverse oflaplace ((2))/(s^2+5s+3) is 4/(sqrt(13))e^{-(5t)/2}sinh((sqrt(13)t)/2)

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inverse oflaplace ((2))/(s^2+5s+3)

inverse laplace

\frac{( 2 )}{s ^{2} + 5 s + 3}

\frac{4}{13} e^{- \frac{5 t}{2}} sinh (\frac{13 t}{2})

Solution steps

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

Expand

\frac{2}{s ^{2} + 5 s + 3} : 2 \cdot \frac{1}{( s + \frac{5}{2} ) ^{2} - \frac{13}{4}}

= L^{- 1} {2 \cdot \frac{1}{( s + \frac{5}{2} ) ^{2} - \frac{13}{4}}}

Use the constant multiplication property of Inverse Laplace Transform:

For function

f (t)

and constant

a : L^{- 1} {a \cdot f (t)} = a \cdot L^{- 1} {f (t)}

= 2 L^{- 1} {\frac{1}{( s + \frac{5}{2} ) ^{2} - \frac{13}{4}}}

L^{- 1} {\frac{1}{( s + \frac{5}{2} ) ^{2} - \frac{13}{4}}} : e^{- \frac{5 t}{2}} \frac{2}{13} sinh (\frac{13 t}{2})

= 2 e^{- \frac{5 t}{2}} \frac{2}{13} sinh (\frac{13 t}{2})

Refine

2 e^{- \frac{5 t}{2}} \frac{2}{13} sinh (\frac{13 t}{2}) : \frac{4}{13} e^{- \frac{5 t}{2}} sinh (\frac{13 t}{2})

= \frac{4}{13} e^{- \frac{5 t}{2}} sinh (\frac{13 t}{2})

\frac{d}{d x} (a x + 4)

\frac{\partial}{\partial y} (\frac{y}{3})

\int \frac{x}{5 - 3 x ^{2}} d x

x d x + \frac{1}{y} d y = 0

\int (e^{- x} - e^{5}) (e^{x} + e^{- 2}) d x

What is inverse oflaplace ((2))/(s^2+5s+3) ?
The answer to inverse oflaplace ((2))/(s^2+5s+3) is 4/(sqrt(13))e^{-(5t)/2}sinh((sqrt(13)t)/2)