The solution of the differential equation $${{{d^2}y} \over {d{x^2}}} = 0$$ with boundary conditions
(i) $${{dy} \over {dx}} = 1$$ at $$x=0$$
(ii) $${{dy} \over {dx}} = 1$$ at $$x=1$$

During the numerical solution of a first order differential equation using the Euler (also known as Euler Cauchy) method with step size $$h,$$ the local truncation error is of the order of

Consider steady flow of water in a situation where two pipe lines (pipe $$1$$ and pipe $$2$$) combine into a single pipe line $$(pipe-3)$$ as shown in the figure. The cross-sectional areas of all three pipelines are constant. The following data is given :

Assuming the water properties and the velocities to be uniform across the cross section of the inlets and the outlet, the exit velocity (in $$m/s$$) in pipe $$3$$ is

The pressure drop for laminar flow of a liquid in a smooth pipe at normal temperature and pressure is