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Exercise 1 (23 marks). Consider a first order quasilinear PDE in the standard form A- du(x, y ) du (x , y ) = C, Ox + B- (1) ay 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)) = Flg(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) = CI and g(x, y, u) = C2, C1, C2 ER, contain the characteristics of the PDE described by da du dT = A, dy = B, dT dT = C. 1. [11 marks] Show by direct substitution that u-y = Fx+- log(u) ) is a solution to yux + uzuy = u , for a suitably well-behaved function F. 2. [12 marks] Demonstrate by substitution in general that Equation (2) is a solution to (1)