Q.1 Use Reduction of order method to find the second solution of dx 4x dy dx #6y=0if y = x + x is the first solution. Q.2 (1) parameters. (ii) Q. 3 (1) (ii) Solve the differential equation Solve the Cauchy-Euler equation.x Solve the differential equation d'y dy -2 + y = dx dx Find the general solution of 2 dy dy +9x- dx dx d^y 16 +24- dx-4 d y dy -16- 5 dxs dx dy dx : 0. et 1 + x = -20y: by the Variation of 5 +9y = 0. d'y dx dx dy Q. 4 (1) Solve the differential equation +9y=6x +2-12e by undetermined dy coefficients (UC) method and verify particular integral by inverse operator method. (ii) Find a second solution of the equation x -x- +2y=0 if y=xsin (In x) is the first dx solution of the equation. dx Q. 5 A vertical beam of light passes through a transparent substance, the rate at which its intensity I decreases is proportional to I(t). In clear sea water the intensity 3 feet below the surface is 25% of the initial intensity I, of the incident beam. What is the intensity of the beam 15 feet below the surface? Here is the depth of the sea in feet. Model the differential equation of this problem using some physical law and hence determine the required.