Unformatted text preview: 1 Solutions 12. The right hand side is . Then 39 . Choose a rectangle
that lies above the -axis. Then both
are continuous on . Picard’s theorem applies and we can conclude there
is a unique solution on an interval about .
13. The right hand side is . Then . Choose a rectangle about
that contains no points on the line
Then both and
are continuous on . Picard’s theorem applies and
we can conclude there is a unique solution on an interval about .
14. The right hand side is
condition , which is not deﬁned at the initial
. Thus Picard’s theorem does not apply. 15. The corresponding integral equation is
have We thus .
. We can write . We recognize this sum as the ﬁrst terms of the Taylor series expansion for
. Thus the limiting function
. It is straightforward to verify that it is a
. Both and
on the whole
-plane. By Picard’s theorem, Theorem 5,
the only solution to the given initial value problem.
16. 1. The equation is separable so separate the variables to get
and the initial condition
implies that the integration constant
, so that the exact solution
is 2. To apply Picard’s method, let and deﬁne ...
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- Fall '08
- Derivative, Constant of integration, Picard