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 ___________.

Given the differential equation $${y^1} = x - y$$ with initial condition $$y(0)=0.$$ The value of $$y(0.1)$$ calculated numerically upto the third place of decimal by the $${2^{nd}}$$ order Runge-Kutta method with step size $$h=0.1$$ is

The function $$f\left( {x,y} \right) = {x^2}y - 3xy + 2y + x$$ has

$$\mathop {Lim}\limits_{x \to 0} {{x\left( {{e^x} - 1} \right) + 2\left( {\cos x - 1} \right)} \over {x\left( {1 - \cos x} \right)}} = \_\_\_\_\_\_.$$