Laplace transform of $${\left( {a + bt} \right)^2}$$ where $$'a'$$ and $$'b'$$ are constants is given by:

Solve the initial value problem
$${{{d^2}y} \over {d{x^2}}} - 4{{dy} \over {dx}} + 3y = 0$$ with $$y=3$$ and
$${{dy} \over {dx}} = 7$$ at $$x=0$$ using the laplace transform technique?

If $$f(t)$$ is a finite and continuous Function for $$t \ge 0$$ the laplace transformation is given by
$$F = \int\limits_0^\infty {{e^{ - st}}\,\,f\left( t \right)dt,} $$ then for $$f(t)=cos$$ $$h$$ $$mt,$$ the laplace transformation is ___________.

The laplace transform of the periodic function $$f(t)$$ described by the curve below
$$i.e.\,\,f\left( t \right) = \left\{ {\matrix{ {\sin \,t,} & {if\left( {2n - 1} \right)\pi < t < 2n\pi \left( {n = 1,2,3,...} \right)} \cr 0 & {otherwise\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \cr } } \right.$$ is ___________.