1
1.(a) Classify the following second order partial differential equation
x2uxx− 2xyuxy + y2uyy + xux + yuy = 0,
where u is a function of two variables x and y. [2 marks]
(b) Find the canonical form of this equation. Be sure to show the change of coordinatesthat
reduces the pde to canonical form. [18 marks]
(c) Hence, solve this to find the general solution of the equation. [5 marks] 2. Let
u(x,y) be a real-valued function satisfying the following partial differential equation
, (1)
for all x,y with 0 ≤ x ≤ π and y ≥ 1. Also suppose that u satisfies the following boundary
conditions
u(0,y) = u(π,y) = 0, for all y ≥ 1 u(x,1) =
f(x), for all x ∈ [0,π]
(a) Assume that u can be written as u(x,y) = X(x)Y (y) rewrite PDE (1) so that you obtain
X00 = λX (2)
and an ODE for Y , where λ is an unknown constant. [3 marks]
(b) Show that if u is not trivial, then the boundary conditions given above lead to
homogeneous boundary conditions for ODE (2). [3 marks]
(c) Show that if λ ≥ 0, then X must be trivial. [4 marks]
(d) Solve the eigenvalue problem for X, i.e. find all the eigenvalues λn and corresponding
eigenfunctions Xn. [3 marks]
(e) For every λn find the corresponding solution Yn to the other ODE from part (a).
[4 marks]
(f) Write down the general solution to (1) and with boundary condition [3 marks]
(g) Find the function value u(π/4,4) in the case that
f(x) = 2sin(x) + 11sin(4x) [5 marks]2
3. (a) Show that the eigenvalue problem
, (3)
has eigenvalues λn = n2π2 + 3 and eigenfunctions yn(x) = e−2x sin(nπx) where
n = 1,2,3,… [ 8 marks]
(b) Put equation (3) into Sturm-Liouville form and thus identify the weight functionw(x).
[5 marks]
(c) Verify that the set of eigenfunctions {yn : n ∈N} is orthogonal on [0;1] with respect to
the weight function w.
Hint: You may find the identity: sin( useful]
[8 marks]
(d) Define the function f : R→ [−1,1] by f(x) = cos(x). Write down an expression for an such
that
(4)
in terms of the relevant integrals. Note: You are not asked to calculate these integrals
or an here, but you should simplify the integrands. [4 marks]
4. (a) Find the general solution to the homogeneous Cauchy-Euler equation
, (5)
[Hint: Look for solutions of the form u = xm.] [5 marks]
(b) By using your answer to part (4a) to find the Green’s function for
,
show that the solution to this inhomogeneous boundary value problem is
[15 marks]3
(c) Hence find the solution to this problem when f(x) = 1.
Simplify your answer as far as possible. [5 marks
