The solution to $$\,\,{x^2}{y^{11}} + x{y^1} - y = 0\,\,$$ is

In numerical integration using Simpson's rule, the approximating function in the interval is a

In a vertical piston-cylinder arrangement the force applied to the piston, pushes water through a nozzle as shown in the figure. The water flows out from the nozzle, and reaches the top of its trajectory. The kinetic and pressure energies at points $$(1), (2)$$ and $$(3),$$ respectively, are

In a $$1$$ $$m$$ thick wall, the temperature distribution at a given instant is
$$T\left( x \right) = {c_0} + {c_1}x + {c_2}{x^2}$$ where $$T$$ is in $${}^ \circ C$$ and $$x$$ is in $$m.$$ The constants are: $${c_0} = {800^ \circ }C,$$ $${c_1} = - {250^ \circ }C/m$$ and $${c_2} = - {40^ \circ }C/{m^2}.$$ The thermal conductivity of the wall is $$50$$ $$W/mK$$ and wall area is $$5\,\,{m^2}.$$ If there is a heat source generating uniform volumetric heating at the rate of $$500$$ $$W/{m^3}$$ inside the wall, then the rate of change of energy storage in the wall, in $$kW,$$ is _____________.