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

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

Temporary hardness in water is caused by the presence of

Principle involved in the relationship between submerged unit weight and saturated weight of a soil is based on