MATC46 Assignment 1, due Jan 24
1. Laplace equation
Verify that the function u(x, y) = ln ^x2 + y2 satisies the Laplace equation u xx + ugg = 0 for all (x, y) 
(0, 0).
2. Inhomogenous constant coe田cient equation
Solve the equation 2ux + 3ug = 4u.
3. Method of characteristics
i. Solve the equation yu x + xu g = 0 with u(0, y) = e-g2 .
ii. In which region of the plane is the solution uniquely determined?
4. An ill-posed problem
ux + yu g = 0
with the boundary condition u(x, 0) = φ(x).
i. For φ(x) = x, show that no solution exists.
ii. For φ(x) = 1, show that there are many solutions.
5. Deriving a PDE model
A lexible chain of length ` is hanging from one end x = 0 but oscillates horizontally. Let the x-axis point downward and the u-axis point to the right. Assume that the force of gravity at each point of the chain equals the weight of the part of the chain below the point and is directed tangentially along the chain. Assume that the oscillations are small. Find the PDE satisied by the chain.
6. Schr
dinger equation
Consider a solution u = u(x,y, z, t) of the Schrodinger equation
iut = -Δu + Vu.
Here Δ = ∂x(2) + ∂g(2) + ∂z(2) denotes the Laplacian, and V = V (x,y, z) is a given real-valued function called the potential. Show that if 1R3 |u|2 = 1 at t = 0, then1R3 |u|2 = 1 for all t. (You may assume that all quantities involved are su伍ciently smooth and decay sufficiently fast at ininity).
We will randomly select 3 questions, for which you will receive points p1 , p2 , p3 E {0, 1, 2, 3} depending on how well you solved them. Let s be the number of questions that you skipped. The total number of points you receive for this assignment is max(p1 + p2 + p3 - s, 0) E {0, 1, . . . , 9}.