Partial Dierential Equations Math 442 C13/C14 Fall 2009 Homework 2 due September 18
1. (Strauss 2.1.1.) Solve utt = c2 uxx , u(x, 0) = ex , ut (x, 0) = sin x. Solution: We use dAlemberts formula with (x) = ex , Then we have
x+ct
(x) = sin x.
sin s ds = c
Partial Dierential Equations Math 442 C13/C14 Fall 2009 Homework 1 Solutions
1. Determine which of the following operators are linear: (a) Lu = uxx + uxy (b) Lu = uux (c) Lu = 4x2 uy 4y 2 uyy Solution: compute: (a) L(u + v ) = (u + v )xx + (u + v )xy = ux
Partial Dierential Equations Math 442 C13/C14 Fall 2009 Homework 6 Solutions
1. Let D = cfw_(x, y ) : x2 + y 2 < 4, and solve u = 0, in D, u = 3 2 cos ,
r = 2.
Solution: Using the formula generated in class, we have that u(r, ) = A0 + rn (An cos(n) + Bn s
Partial Dierential Equations Math 442 C13/C14 Fall 2009 Homework 5 Solutions
1. (Strauss 5.2.2.) Show that cos(x) + cos(x) is periodic if is a rational number and compute its period. What happens if is not rational? Solution: Let us rst notice that if f,
Partial Dierential Equations Math 442 C13/C14 Fall 2009 Homework 4 due October 9
1. Consider the boundary value problem A + A = 0, A (0) + aA(0) = 0, A(L) = 0.
(a) Show that if a < 0, then there is no negative eigenvalue. (b) Under which conditions is the
Partial Dierential Equations Math 442 C13/C14 Fall 2009 Homework 3 Solutions
1. Here we will prove that solutions to the heat equation satisfy (some of) the invariance principles mentioned in class, or in the book in 2.4. That is, if u(x, t) is a solution