9. Show that the initial value problem
u
t
+
u
x
=
x
u
(
x, x
) = 1
has no solution. Give a reason for the problem.
Note that the initial line in this case is NOT the
x
axis (
t
= 0) but the line
t
=
x
.
The ODEs are
dx
dt
= 1
,
du
dt
=
x
and the initial condition for each
x
(
t
=
x
) =
ξ,
u
(
x
(
t
=
x
)
, t
=
x
) = 1
Solve the characteristic equation
x
=
t
+
K
and use the initial condition
ξ
=
ξ
+
K
so
K
= 0 and the characteristic is
x
=
t
This is the only line, there is no family as in other problems.
Substitute in the
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 Fall '08
 BELL,D
 Constant of integration, Boundary value problem, initial condition

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