Math 425 - Notes and Homework Thursday January 21, 2010

Topics for this week -

1. Basic PDE concepts

Examples -

1. The "big three" (or four) PDEs - Laplace, heat, wave (and transport) equations.
2. Methods for finding solutions - first-order equations like 3u_x + 4u_y = Q, where Q is zero, or u, or u + f(x,y). Initial-value problems for this.
3. Initial, boundary and initial/boundary value problems for second order equations.

Second Homework Assignment - due Thursday, January 28

1. Reading: Read the notes on vecctor fields/first-order equations
2. 1. Find the general solution of u_xy = x²y for the function u(x,y).
   2. Ditto for: yu_xy + 2u_x = x (Hint: first integrate with respect to x)
   3. For the preceding PDE, find the solution that satisfies u(x,1) = 0 and u(0,y) = 0.
   4. Solve: u_x + 2u_y = 0, 5u_x + 6u_y = 0, cu_x + du_y = 0. (These are three separate problems)
   5. Solve the equation yu_x + xu_y = 0 with u(0,y) = exp(-y²). In which region of the xy plane is the solution uniquely determined?
   6. Solve u_x + u_y + u = e^x+2y with u(x,0) = 0.
   7. Solve the equation u_x + 2u_y + (2x - y)u = 2x² + 3xy - 2y².

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