In lectures, we have discussed first order quasilinear PDEs.That is, PDEs of the general form A(x, y, u) ?u(x, y) ?x + B(x, y,u) ?u(x, y) ?y = C(x, y, u), (1) for some A, B and C. To solve suchPDEs we first find characteristics, curves in the solution space(x, y, u) parametrically given by (x(? ), y(? ), u(? )), whichsatisfy dx d? = A(x, y, u), dy d? = b(x, y, u), du d? = C(x, y, u).(2) We find solutions to these equations in the form f(x, y, u) =C1 and g(x, y, u) = C2 where C1 and C2 are arbitrary constants. Theindependent functions f and g are then used to write the generalsolution to Equation (1) f(x, y, u(x, y)) = F [g(x, y, u(x, y))] ,(3) where F is a sufficiently smooth function (that is, you canexpect in this question that its derivative exists everywhere). 1.[12 marks] Verify that (3) for any suitable F and for any f and gas described above is actually a solution to the PDE (1). That is,you should show that given (2) which describe the functions f and gand the solution (3), then Equation (1) is always satisfied. HINT:This is not as simple as it sounds. You should first attempt todifferentiate f(x, y, u) = C1 and g(x, y, u) = C2 by ? anddifferentiate the solution (3) first with respect to x and thenwith respect to y and use the resultant simultaneous equations todeduce (1).
