Unformatted text preview: 1/2. Now we can use partial integration:
u=
ˆ ⌘
+ g (⇠ )
2 y
+ g x2 + y 3
2
(You can easily verify that this solves the DE, for any diﬀerentiable function
g ).
=) u= All that remains is to apply the initial condition:
y
y
u(0, y ) = , =)
=
2
2
=) 1 y
+ g y 3 =) g y 3 = y
2
p
g (y ) = 3 y Therefore the solution to the given BVP is
y p2
u(x, y ) =
+ 3 x + y3.
2 The transformation ⇠ = x, ⌘ = x2 + y 3 also works, but is a bit more diﬃcult. AMATH 350 Assignment #7 Solutions  Fall 2013 Page 10 BONUS: If we try to satisfy the condition u (x, 0) = x, we obtain the expression
x = g (x2 ). If our domain were restricted to x 0, we could conclude that
p
p
g (x) = x, while if we had x < 0 we could conclude that g (x) =
x,
but we cannot have both of these at once.
The real reason becomes apparent if we plot the characteristics. The
curves x2 + y 3 = C look like this: Recall that for this kind of problem, information from the initial condition
is transported along the characteristics. Because the characteristics loop
around in the 1st and 2nd quadrants, if the initial condition were given
along the positive xaxis, that would actually determine the values of u
along the negative xaxis as well. In other words, we will only be able
to satisfy an initial condition given along the xaxis if the values of u for
negative values of x are consistent with the values of u for positive values
of x.
If f (x) were even, we’d be ok....
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 Fall '12
 DavidHamsworth
 Trigraph, Emoticon, ASCII art, BVP

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