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1. (i) Which of the following differential operators 4 are linear? Justify briefly your answers (b) (ii) Determine the values of the parameters a,b,c,d R for which the equation 16a,2 + bu, +cu,+du (1) is parabolic. (iii) (a) Find the general solution u(x,y) of the equation uzy=0. (2) (b) Find particular solution of equation (2) which satisfies Total marks 15 the conditions u(z,x) -0 and u(x,0)= 2. Consider the initial value problem for the homogeneous wave equa- tion rer (3) (i) State the d' Alembert formula for the solution of (3) (ii) Solve initial value problem (3) with g(x) e . (iii) Let u be the solution to (3) constructed in part (ii) and -3t Find the limit um u(x,t) 3. Consider the initial value problem for the heat equation rea (4) (i) State the formila for the solution of (4) in the case when f the Heaviside step function f(x) (3)={:, (5) (ii) Use the formula in part (i) to find the solution of (4) in the case when f(x)= (6) 4. (i) Find all positive values of for which the problem X'(0) = X'(r/2) = 0, (7) has non- trivial solutions (i.e. x 0). Find these solutions the result ofpart (i) tosolve the following initial bound- ary value problem by the method of separation of variables O<1<1/2,1>0, u(z,0)=1 cos(2x) cos(4x), O(8) vz(O,L) t>0. (iii) Let be the solution to (8) constructed in (ii). Find the limit u(z.t) for all values 5. Let {1+y2< 1} be the open unit disk in R² (i) Find the values oi abeR for which the function is super- harmonic in 2. (ii) State the maximum and the minimum principle for har- monic functions 2. (iii) Prove using the minimum principle that the function is not harmonic 2. (iv) Using the maximum and the minimum principle, prove the uniqueness theorem for the Dirichlet problem in 9: { (9) (v) Find the solution of the Dirichlet problem (9).

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