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Math Help - Initial Value Problem: PLEASE HELP!

  1. #1
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    Initial Value Problem: PLEASE HELP!

    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!
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  2. #2
    Super Member Rebesques's Avatar
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    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.
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