Math 3326
Spring Semester 2009
Problem Set #1
1. Show that if pis a continuously di¤erentiable function of one variable, the
…rst-order partial di¤erential equation
ut=p(u)ux
has a solution implicitly de…ned by
u(x; t) = '(x+p(u)t);
in which 'can be any continuously di¤erentiable function of one variable.
Use this idea to determines (perhaps implicitly) a solution of each of the
following equations:
(a) ut=kux(with kbeing a nonzero constant)
(b) ut=uux
(c) ut= cos(u)ux
(d) ut=usin(u)ux:
2. Show that u(x; y) = ln((xx0)2+ (yy0)2)satis…es uxx +uyy = 0 for
all pairs (x; y)of real numbers except (x0; y0):
3. Let kbe a positive constant. Let
u(x; t) = 1
2pkt Z1
1
e(x)2=4ktf()d;
in which fis continuous on the real line. Show that ut=kuxx for 1 <
x < 1,t > 0:Also determine u(x; t)when f(x) = 1 for all real x: Hint:
Use a change of variable and the standard result that
Z1
1
ew2
dw =p:
1