What is laplacetransform xe^x ?

The answer to laplacetransform xe^x is 1/((s-1)^2)

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laplacetransform xe^x

laplace transform

x e^{x}

\frac{1}{( s - 1 ) ^{2}}

Solution steps

L {e^{x} x}

Use Laplace Transform table:

L {t^{k} f (t)} = (- 1)^{k} \frac{d ^{k}}{d s ^{k}} (L {f (t)})

For

x e^{x} : f (x) = e^{x}, k = 1

= (- 1)^{1} \frac{d}{d s} (L {e^{x}})

L {e^{x}} : \frac{1}{s - 1}

\frac{d}{d s} (\frac{1}{s - 1}) = - \frac{1}{( s - 1 ) ^{2}}

= (- 1)^{1} (- \frac{1}{( s - 1 ) ^{2}})

= \frac{1}{( s - 1 ) ^{2}}

x y^{^{'}} + y = - 5 x y^{2}, y (1) = 3

y^{^{''}} + 2 y^{^{'}} = 2 t + 3

\int_{1}^{4} \frac{7}{x} d x

\frac{60 t ^{2}}{t ^{2} + 150}

\frac{\partial}{\partial x} (8 x y - \frac{1}{4} (x + y)^{4})

What is laplacetransform xe^x ?
The answer to laplacetransform xe^x is 1/((s-1)^2)