a)Solve the following PDE by separation of variables method , 3du/dx + 2du/dy = 0 , u(x,0) = 4e^{^-x}

(5 marks)

b)Solve the given equations;

i) dx/xy=dy/y^2= dz/zxy-2x^2

(5 Marks)

c)Obtain the partial differential equation whose solutions is given by F(u,v) = 0 , where u and v are functions of x, y and z, while z is a function of x and y.(10 Marks)

d)Show that the simultaneous equations:

p1dx + Q1dy + R1dt = 0

p2dx + Q2dy + R2dt = 0

Can be reduced to take the form of:dx/p = dy/Q = dz/R

Hence solve: dx/x +z =dy/y = dt/ z + y^2

e)Find the Orthogonal Trajectories of the surface, x^{^2} +y^2 + z^2 = a^2 with its intersection with the family of surface. xy = cz. (10 Marks)

f)Solve the heat equation , du/dt - d^{^2}u/dx^{^2 = 0}by the method of separation of variables, for u(0,t) = u(4,t)=0 and u(x,0) =x-4.(10 Marks