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Math Help - how to find a general solution (dfq)

  1. #1
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    how to find a general solution (dfq)

    ok. so i am given an equation.
    y''+4y=12x
    i am told that the a particular equation is : y=3x
    i understand that, it makes the first function =0. i know how to do that.
    but then i am given the complimentary function as y=c1cos(2x)+c2sin(2x)

    so my question is, where does that come from? how could i have found that on my own? i have a test and i believe our professor mentioned that we wouldn't be given those as we have been in our homework sets, so i need to be capable of finding it just from
    y''+4y=12x.

    i don't know how though. i did figure the particular one out, but this i am stuck.
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  2. #2
    A Plied Mathematician
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    Complimentary functions are solutions of the corresponding homogeneous equation y'' + 4y = 0. Generally, you find solutions to second-order linear constant coefficient ode's like this by assuming a solution of the form y=Ae^{rx}. You plug that solution in, and you'll get an equation for r that, in this case, will have complex solutions. Use Euler's formula to get the sines and cosines out of the exponentials.

    Make sense?
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  3. #3
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    yeah that makes sense, thanks. i have to run off to work unfortunately but i'll practice with it when i get back. but i'm pretty sure i understand. thank you
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  4. #4
    A Plied Mathematician
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    Ok, let me know how it goes.
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