Exercise 1 (23 marks). Consider a first order quasilinear PDE in the standard form du(x, y) du(x, y) + B = C. ay (1) where A, B and C are some well-behaved functions of x, y and u(x, y). The general solution of this PDE has been shown in lectures as taking the form f(x, y, u(x, y)) = F[g(x, y, u(x, y))] , (2) where F is a suitable arbitrary function and the level surfaces of the functions f and g, f(x, y, u) = C1 and g(x, y, u) = C2, C1, C2 ER, contain the characteristics of the PDE described by dx dy du dT = A, dT = B, = C. 1. [11 marks] Show by direct substitution that u-y = Fx+ - log(u) ) u is a solution to yur + uly = u2, for a suitably well-behaved function F. 2. [12 marks] Demonstrate by substitution in general that Equation (2) is a solution to (1)