Consider the differential equation $${{dy} \over {dx}} = \left( {1 + {y^2}} \right)x\,\,.$$ The general solution with constant $$'C'$$ is

The integral $$\,\int\limits_1^3 {{1 \over x}\,\,dx\,\,\,} $$ when evaluated by using simpson's $$1/{3^{rd}}$$ rule on two equal sub intervals each of length $$1,$$ equals to

A stone with mass of $$0.1$$kg is catapulted as shown in the figure. The total force $${F_x}$$ (in N) exerted by the rubber band as a function of distance $$x$$ (in m) is given by $${F_x} = 300{x^2}.$$ If the stone is displayed by $$0.1$$ m from the un-stretched position (x = 0) of the rubber band, the energy stored in the rubber bad is

A 1 kg block is resting on a surface with coefficient of friction $$\mu $$ $$=0.1.$$ A force of $$0.8$$ $$N$$ is applied to the block as shown in the figure. The friction force is