What is inverse oflaplace (45)/(s(s+3)^2) ?

The answer to inverse oflaplace (45)/(s(s+3)^2) is 5\H(t)-5e^{-3t}-e^{-3t}*15t

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inverse oflaplace (45)/(s(s+3)^2)

inverse laplace

\frac{45}{s ( s + 3 ) ^{2}}

5 H (t) - 5 e^{- 3 t} - e^{- 3 t} \cdot 15 t

Solution steps

L^{- 1} {\frac{45}{s ( s + 3 ) ^{2}}}

Take the partial fraction of

\frac{45}{s ( s + 3 ) ^{2}} : \frac{5}{s} - \frac{5}{s + 3} - \frac{15}{( s + 3 ) ^{2}}

= L^{- 1} {\frac{5}{s} - \frac{5}{s + 3} - \frac{15}{( s + 3 ) ^{2}}}

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{5}{s + 3}} - L^{- 1} {\frac{15}{( s + 3 ) ^{2}}}

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

L^{- 1} {\frac{5}{s + 3}} : 5 e^{- 3 t}

L^{- 1} {\frac{15}{( s + 3 ) ^{2}}} : e^{- 3 t} \cdot 15 t

= 5 H (t) - 5 e^{- 3 t} - e^{- 3 t} \cdot 15 t

7 x e^{- k x}

\int \frac{54}{1 + 36 x ^{2}} d x

f (x) = arcsin (\frac{x}{2})

x \to 0 + lim (ln (x) + \frac{1}{x ^{2}})

\int sec (14 w) tan (14 w) d w

What is inverse oflaplace (45)/(s(s+3)^2) ?
The answer to inverse oflaplace (45)/(s(s+3)^2) is 5\H(t)-5e^{-3t}-e^{-3t}*15t