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

The equation $$\,\,\,{{{d^2}u} \over {d{x^2}}} + \left( {{x^2} + 4x} \right){{dy} \over {dx}} + y = {x^8} - 8\,\,{u \over {{x^2}}} = 8.\,\,\,$$ is a

Four arbitrary points $$\,\,\,\left( {{x_1},{y_1}} \right),\,\,\left( {{x_2},{y_2}} \right),\,\,\left( {{x_3},{y_3}} \right),\,\,\left( {{x_4},{y_4}} \right),\,\,\,\,$$ are given in the $$xy$$ $$-$$ plane using the method of least squares. If regression of $$y$$ upon $$x$$ gives the fitted line $$y=ax+b;$$ and regression of $$x$$ upon $$y$$ gives the fitted line $$x=cy+d,$$ then

Value of the function $$\mathop {Lim}\limits_{x \to a} \,{\left( {x - a} \right)^{x - a}}$$ is _______.