Unformatted text preview: c. The set of ODEs is du
=u
dt dx
= −t2
dt The solution of the ﬁrst is
x = x0 − 13
t
3 Now integrate the second ODE
ln u(x, t) = −t + ln K
or u(x, t) = K e−t At t = 0 the solution is
u(x0 , 0) = f (x0 ) = K plug t = 0 in the solution u Thus when substituting for x0 in the solution
u(x, t) = e−t f x + d. The set of ODEs is 13
t
3 du
=t
dt dx
=x
dt The solution of the ﬁrst is
ln x = ln x0 + t
or
x = x0 et
Now integrate the second ODE
u(x, t) = 12
t +K
2 At t = 0 the solution is
u(x0 , 0) = f (x0 ) = K plug t = 0 in the solution u Thus when substituting for x0 in the solution
u(x, t) = 12
t + f x e−t
2 252 ...
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 Fall '08
 BELL,D
 dt

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