What is laplacetransform t^3e^{-2t}-3cos(5t) ?

The answer to laplacetransform t^3e^{-2t}-3cos(5t) is 6/((s+2)^4)-(3s)/(s^2+25)

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laplacetransform t^3e^{-2t}-3cos(5t)

laplace transform

t^{3} e^{- 2 t} - 3 cos (5 t)

\frac{6}{( s + 2 ) ^{4}} - \frac{3 s}{s ^{2} + 25}

Solution steps

L {e^{- 2 t} t^{3} - 3 cos (5 t)}

Use the linearity property of Laplace Transform:

For functions

f (t), g (t)

and constants

a, b : L {a \cdot f (t) + b \cdot g (t)} = a \cdot L {f (t)} + b \cdot L {g (t)}

= L {t^{3} e^{- 2 t}} - 3 L {cos (5 t)}

L {t^{3} e^{- 2 t}} : \frac{6}{( s + 2 ) ^{4}}

L {cos (5 t)} : \frac{s}{s ^{2} + 25}

= \frac{6}{( s + 2 ) ^{4}} - 3 \cdot \frac{s}{s ^{2} + 25}

Refine

\frac{6}{( s + 2 ) ^{4}} - 3 \frac{s}{s ^{2} + 25} : \frac{6}{( s + 2 ) ^{4}} - \frac{3 s}{s ^{2} + 25}

= \frac{6}{( s + 2 ) ^{4}} - \frac{3 s}{s ^{2} + 25}

What is laplacetransform t^3e^{-2t}-3cos(5t) ?
The answer to laplacetransform t^3e^{-2t}-3cos(5t) is 6/((s+2)^4)-(3s)/(s^2+25)