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!

2. 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 calculate $c_1=y'(0), c_2=y''(0),...$ and express the solution in terms of a Taylor expansion: $y(x)=12+c_1x+...$
As is well known, this converges in the neighbourhood $(-r,r)$ of zero, where $r=({\rm limsup}_n (|c_n|/n!)^{1/n})^{-1}$

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.