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Math Help - Simple equation - question

  1. #1
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    Simple equation - question

    Hi!

    I just started up some repetition but got stuck when I looked at my techers answers.

    I have this equation: x + (k/m)x = 0

    Then I get from the auxilliary solution that: x = Acos(sqrt(k/m))

    However my teacher has written: x = Acos(sqrt(k/m)+a)

    So he has some phase-constant a, and here is my question.

    I see that this isnt that strange but how did it enter? Can someone show it to mathematically? Logically I think it makes sence but I'd like to see how.

    Thanks for answers.
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  2. #2
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    What variable is x a function of? x can not possibly equal A\cos{\left(\sqrt{\frac{k}{m}}\right)} since this is a number, not a function.
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  3. #3
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    x is a function of time, because the DE is the equation of motion for a mass on a spring undergoing simple harmonic motion. There should be t's multiplying the square roots inside the trig functions.

    I don't know what your "auxiliary" solution is; however, without initial conditions, it is incorrect. Your teacher's answer is correct. You can either write the homogeneous solution as

    x(t) = A \cos(t\sqrt{k/m} + a), with the phase angle, or as
    x(t) = A \cos(t\sqrt{k/m}) + B \sin(t\sqrt{k/m}).

    It can be shown that these two approaches are equivalent.

    Incidentally, I should point out that in the physics world, at least, there is a convention that primes (r'') are usually reserved for spatial derivatives, and dots (\ddot{r}) are usually reserved for temporal derivatives.
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  4. #4
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    Thanks for the answers Ackbeet! Yes its true I forgot to write the t inte my equation.

    Is there anyone who would like to show,mathematically, that these 2 statements are the same?

    , with the phase angle, or as
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  5. #5
    A Plied Mathematician
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    Use the addition of angles formula for the cosine. You'll absorb the cos(a) and sin(a) into the A and B constants of integration, and then you're done. Going the other way, you can simply re-define the A and B of the second equation to have the cos(a) and sin(a) that you need, use the addition of angles formula and simplify into the first equation. Incidentally, the A in the second equation is not the same A as in the first equation. You just need to have two constants (the two constants of integration you should always get for a second-order ODE).

    It all comes down to the addition of angles formula for the cosine.
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  6. #6
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    Thanks for answer again Ackbeet. Very simple when you know what to do.
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  7. #7
    A Plied Mathematician
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    You're very welcome. Have a good one!
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