What is laplacetransform sin(2t)sin(5t) ?

The answer to laplacetransform sin(2t)sin(5t) is (20s)/((s^2+9)(s^2+49))

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laplacetransform sin(2t)sin(5t)

laplace transform

sin (2 t) sin (5 t)

\frac{20 s}{( s ^{2} + 9 ) ( s ^{2} + 49 )}

Solution steps

L {sin (2 t) sin (5 t)}

Rewrite using trig identities

L {- \frac{1}{2} cos (7 t) + \frac{1}{2} cos (- 3 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 {cos (7 t)} + \frac{1}{2} L {cos (- 3 t)}

L {cos (7 t)} : \frac{s}{s ^{2} + 49}

L {cos (- 3 t)} : \frac{s}{s ^{2} + 9}

= - \frac{1}{2} \cdot \frac{s}{s ^{2} + 49} + \frac{1}{2} \cdot \frac{s}{s ^{2} + 9}

Simplify

- \frac{1}{2} \frac{s}{s ^{2} + 49} + \frac{1}{2} \frac{s}{s ^{2} + 9} : \frac{20 s}{( s ^{2} + 9 ) ( s ^{2} + 49 )}

= \frac{20 s}{( s ^{2} + 9 ) ( s ^{2} + 49 )}

\frac{\partial}{\partial x} (y x + 2 x)

\int \frac{1 + sin ( x )}{1 - sin ( x )} d x

\int x ln (x + 2) d x

\int_{0}^{1} 2 π x e^{- x^{2}} d x

\frac{d}{d x} (x y^{^{'}} - y)

What is laplacetransform sin(2t)sin(5t) ?
The answer to laplacetransform sin(2t)sin(5t) is (20s)/((s^2+9)(s^2+49))