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Solved Problems

2
(1 + y) (u σσ + u ) − (1 + y) cos(x)u − (1 + y) sin(x)u = 0
σ
τ
ττ
2
σ + τ 2 (u σσ + u ) − σu − τu = 0
τ
ττ
σ
σu + τu τ
σ
u σσ + u ττ =
2
σ + τ 2

Exercise 66. Classify as hyperbolic, parabolic, or elliptic in a region R each of
the equations:

(a) u = (pu )
x x
t
2
(b) u = c u xx − γu
tt
(c) (qu ) + (qu ) = 0
x x
t t
where p(x), c(x, t), q(x, t), and γ(x) are given functions that take on only positive

values in a region R of the (x, t) plane.
Solution:

(a)

u = (pu )
x x
t
pu xx + 0u + 0u + p u − u = 0
xt
x x
tt
t
2
Since 0 − p0 = 0, the equation is parabolic.
(b)

2
u = c u xx − γu
tt
2
u + 0u − c u xx + γu = 0
tx
tt
2
2
Since 0 − (1)(−c ) > 0, the equation is hyperbolic.
(c)

(qu ) + (qu ) = 0
x x
t t
qu xx + 0u + qu + q u + q u = 0
t t
x x
xt
tt
2
Since 0 − qq < 0, the equation is elliptic.

Exercise 67. Transform each of the following equations for φ(x, y) into canonical
form in appropriate regions

2
2
(a) φ xx − y φ + φ − φ + x = 0
yy
x
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