What is laplacetransform 2te^{3t}+4t^2e^{-5t} ?

The answer to laplacetransform 2te^{3t}+4t^2e^{-5t} is 2/((s-3)^2)+8/((s+5)^3)

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laplacetransform 2te^{3t}+4t^2e^{-5t}

laplace transform

2 t e^{3 t} + 4 t^{2} e^{- 5 t}

\frac{2}{( s - 3 ) ^{2}} + \frac{8}{( s + 5 ) ^{3}}

Solution steps

L {2 e^{3 t} t + 4 e^{- 5 t} t^{2}}

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)}

= 2 L {t e^{3 t}} + 4 L {t^{2} e^{- 5 t}}

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

L {t^{2} e^{- 5 t}} : \frac{2}{( s + 5 ) ^{3}}

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

Refine

2 \frac{1}{( s - 3 ) ^{2}} + 4 \frac{2}{( s + 5 ) ^{3}} : \frac{2}{( s - 3 ) ^{2}} + \frac{8}{( s + 5 ) ^{3}}

= \frac{2}{( s - 3 ) ^{2}} + \frac{8}{( s + 5 ) ^{3}}

\frac{d}{d x} (\frac{1}{2 x - 3})

5 - 4 x

e^{x^{5}}

\int sin (x^{2}) d x

\int_{0}^{\infty} e^{- 3 x} d x

What is laplacetransform 2te^{3t}+4t^2e^{-5t} ?
The answer to laplacetransform 2te^{3t}+4t^2e^{-5t} is 2/((s-3)^2)+8/((s+5)^3)