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 solvesuch PDEs we first find characteristics, curves in the solutionspace (x, y, u) parametrically given by (x(? ), y(? ), u(? )),which satisfy 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 arbitraryconstants. The independent functions f and g are then used to writethe general solution 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 can expect in this question that its derivative existseverywhere). 1. [12 marks] Verify that (3) for any suitable F andfor any f and g as described above is actually a solution to thePDE (1). That is, you should show that given (2) which describe thefunctions f and g and the solution (3), then Equation (1) is alwayssatisfied. HINT: This is not as simple as it sounds. You shouldfirst attempt to differentiate f(x, y, u) = C1 and g(x, y, u) = C2by ? and differentiate the solution (3) first with respect to x andthen with respect to y and use the resultant simultaneous equationsto deduce (1). 1 2. [11 marks] Show that the implicit equation u =y + F x + y u ? log (u) for sufficiently smooth arbitrary F is asolution to the PDE y ?u ?x + u 2 ?u ?y = u 2 .

Can u kindly do both of them, or just do question 1,thanks