Consider the initial-value problem:
y' = cos(6t+6sin(y)), y(0) = 12
What can you say, and why, about each of the following?
(a) How many solutions, if any, there are to that initial-value problem.
(b) On what interval such solutions(s), if any, are defined
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I've been working on this problem for hours now and am completely stumped. Any help would be appreciated thanks!
You can use Picard's existence theorem about (a,b)=(0,12).
However, let's use brute force.
We see from the differential equation that y has derivatives of all orders. So we can calculateand express the solution in terms of a Taylor expansion:
=12+c_1x+...)
As is well known, this converges in the neighbourhoodof zero, where
Ps. There is a delicate point involving whether the Taylor series will converge to y. Since y is expressed in terms of the values of the analytic functions sin and cos, there is no problem there.
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