The general solution of the differential equation $${x^2}{{{d^2}y} \over {d{x^2}}} - x{{dy} \over {dx}} + y = 0$$ is

The radial displacement in a rotating disc is governed by the differential equation $$\,\,{{{d^2}u} \over {d{x^2}}} + {1 \over x}{{du} \over {dx}} - {u \over {{x^2}}} = 8x.\,\,\,$$ where $$u$$ is the displacement and $$x$$ is the radius. If $$u=0$$ at $$x=0$$ and $$u=2$$ at $$x=1,$$ calculate the displacement at $$\,x = {1 \over {2.}}$$

A car moving with uniform acceleration covers 450 m in a 5 second interval, and covers 700 m in the next 5 second interval. The acceleration of the car is

A cylinder of mass $$10$$ kg and area of cross-section $$0.1$$ $${m^2}$$ is tied down with string in a vessel containing two liquids as shown in Figure. Calculate gauge pressure on the cylinder bottom and the tension in the string. Density of water $$ = 1000\,\,kg/{m^3}.$$ Specific gravity of $$A=0.8$$. Specific gravity of $$B$$ (water) $$=1.0$$