The Laplace transform of the function
$$\eqalign{ & f\left( t \right) = k,\,0 < t < c \cr & \,\,\,\,\,\,\,\,\, = 0,\,c < t < \infty ,\,\, \cr} $$
is

If $$c$$ is a constant, then the solution of $${{dy} \over {dx}} = 1 + {y^2}$$ is

For the function $$\phi = a{x^2}y - {y^3}$$ to represent the velocity potential of an ideal fluid, $${\nabla ^2}\,\,\phi $$ should be equal to zero. In that case, the value of $$'a'$$ has to be

Limit of the function, $$\mathop {Lim}\limits_{n \to \infty } {n \over {\sqrt {{n^2} + n} }}$$ is _______.