Consider the following second order linear differential equation $${{{d^2}y} \over {d{x^2}}} = - 12{x^2} + 24x - 20$$
The boundary conditions are: at $$x=0, y=5$$ and at $$x=2, y=21$$
The value of $$y$$ at $$x=1$$ is

Water is following at a steady rate through a homogeneous and saturated horizontal soil strip of $$10$$m length. The strip is being subjected to a constant water head $$(H)$$ of $$5$$m at the beginning and $$1$$m at the end. If the governing equation of flow in the soil strip is $$\,\,{{{d^2}H} \over {d{x^2}}} = 0\,\,$$ (where $$x$$ is the distance along the soil strip), the value of $$H$$ (in m) at the middle of the strip is _______.

The solution of the ordinary differential equation $${{dy} \over {dx}} + 2y = 0$$ for the boundary condition, $$y=5$$ at $$x=1$$ is

The solution to the ordinary differential equation $${{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}} - 6y = 0\,\,\,$$ is