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Unformatted text preview: CHAPTER 1 1 Introduction and Applications 1.1 Basic Concepts and Deﬁnitions Problems
1. Give the order of each of the following PDEs
a.
b.
c.
d.
e. uxx + uyy = 0
uxxx + uxy + a(x)uy + log u = f (x, y )
uxxx + uxyyy + a(x)uxxy + u2 = f (x, y )
u uxx + u2 + eu = 0
yy
ux + cuy = d 2. Show that
u(x, t) = cos(x − ct)
is a solution of
ut + cux = 0
3. Which of the following PDEs is linear? quasilinear? nonlinear? If it is linear, state
whether it is homogeneous or not.
a.
b.
c.
d.
e.
f.
g.
h.
i. uxx + uyy − 2u = x2
uxy = u
u ux + x uy = 0
u2 + log u = 2xy
x
uxx − 2uxy + uyy = cos x
ux (1 + uy ) = uxx
(sin ux )ux + uy = ex
2uxx − 4uxy + 2uyy + 3u = 0
ux + ux uy − uxy = 0 4. Find the general solution of
uxy + uy = 0
(Hint: Let v = uy )
5. Show that
is the general solution of y
u = F (xy ) + x G( )
x
x2 uxx − y 2 uyy = 0 1 ...
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.
 Fall '08
 BELL,D

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