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Math Help - Differential equation

  1. #1
    Member kezman's Avatar
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    Differential equation

    Find all the solutions
    y''(x) + y(x) = cos(x) + xy'(0) + y(0)

    I found  \frac{1}{2}x sen(x) as 1 solution.
    I dont know if its the only one.
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  2. #2
    Flow Master
    mr fantastic's Avatar
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    Quote Originally Posted by kezman View Post
    Find all the solutions
    y''(x) + y(x) = cos(x) + xy'(0) + y(0)

    I found  \frac{1}{2}x sen(x) as 1 solution.
    I dont know if its the only one.
    You mean you have found y = \frac{x}{2} \, {\color{red}\sin} x as a particular solution to {\color{red}y''(x) + y(x) = \cos x}.

    You're expected to know that the solution to y''(x) + y(x) = \cos x + xy'(0) + y(0) is the homogeneous solution plus a 'total' particular solution.

    The homogeneous solution is the solution to y''(x) + y(x) = 0 and you're expected to be able to solve this.

    You have a particular corresponding to the \cos x term on the right hand side of the DE. Now you need a particular solution corresponding to the term y'(0) x + y(0). I suggest trying one of the form y = ax + b and getting a and b in terms of y'(0) and y(0). Then the 'total' particular solution is y = \frac{x}{2} \sin x + ax + b.


    If you're stuck on finding the homogeneous solution or the 'total' particular solution, post what you've done and where you get stuck.
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  3. #3
    Super Member wingless's Avatar
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    I think it's "sen" instead of "sin" in Spanish (or maybe some other languages).
    (I'm telling this because you highlighted that sin, mr fantastic ;p)
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