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Math Help - transcendental functions

  1. #1
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    transcendental functions

    Let g be a function everywhere continuous and not identically zero.
    Show that if f'(t) = g(t)f(t) for all real t, then either f is identically zero or f does not take on the value zero.

    Thanks for the help.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    note we have the 1st order differential eqn (y = f(t))

    dy/dt = g(t)y

    Then y = 0 is an equillibrium solution as dy/dt = 0 = g*0= 0

    At this point we could invoke the existence and uniqueness thm
    to conclude any other solution cannot pass through 0 or we can proceed
    directly:

    We can separate the variables


    dy/y = g(t)dt

    Integrating we obtain ln|y| = int(g(t)dt) + C

    Therfore its obvious y cannot take on the value 0 as ln|y| is not defined

    if y = 0
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