PROBLEM Show that if f(x) is a function only of x, then u=f(x) sin (ay + b) is a solution of the partial differential equation (a and b are constants) provided that f(x) satisfies the ordinary differential equation dx2 dx Hence show that u = (A + Bx)e"sin (ay + b) is a solution of the partial differential equation, where A and B are arbitrary constants PROBLEM 2 Convert the three-dimensional Wave equation in terms of (x.yz) into spherical polar coordinates (r, θ, Φ), write down the equation below. PROBLEM 3 A trucking company estimates that the cost of running a truck is 0.02(201+ dollars per mile at a given speed v. The driver earns $20 per hour. Find the cost for a journey of 200 miles. What speed is recommended to minimize the cost PROBLEM 4 A solid body of volume V and surface area S is formed by joining together two cubes of different sizes so that every point on one side of the smaller cube is in contact with the larger cube. If S = 12 m2, find the maximum and/or minimum values of V for which both cubes have non-zero volumes?