What is laplacetransform e^{7t}sin^2(t) ?

The answer to laplacetransform e^{7t}sin^2(t) is 2/((s-7)(s^2-14s+53))

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laplacetransform e^{7t}sin^2(t)

laplace transform

e^{7 t} sin^{2} (t)

\frac{2}{( s - 7 ) ( s ^{2} - 14 s + 53 )}

Solution steps

L {e^{7 t} sin^{2} (t)}

Rewrite using trig identities

L {\frac{1}{2} e^{7 t} - \frac{1}{2} e^{7 t} cos (2 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)}

= \frac{1}{2} L {e^{7 t}} - \frac{1}{2} L {e^{7 t} cos (2 t)}

L {e^{7 t}} : \frac{1}{s - 7}

L {e^{7 t} cos (2 t)} : \frac{s - 7}{( s - 7 ) ^{2} + 4}

= \frac{1}{2} \cdot \frac{1}{s - 7} - \frac{1}{2} \cdot \frac{s - 7}{( s - 7 ) ^{2} + 4}

Simplify

\frac{1}{2} \frac{1}{s - 7} - \frac{1}{2} \frac{s - 7}{( s - 7 ) ^{2} + 4} : \frac{2}{( s - 7 ) ( s ^{2} - 14 s + 53 )}

= \frac{2}{( s - 7 ) ( s ^{2} - 14 s + 53 )}

What is laplacetransform e^{7t}sin^2(t) ?
The answer to laplacetransform e^{7t}sin^2(t) is 2/((s-7)(s^2-14s+53))