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Math Help - clarification with simple differential equation

  1. #1
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    clarification with simple differential equation

    Hi all! My problem concerns the simple differential equation:

    clarification with simple differential equation-codecogseqn-34.gif

    I integrated four times and I ended up with this general solution (c constant):

    clarification with simple differential equation-codecogseqn-35.gif

    I'd assume this is the right solution. I looked up the answer afterwards and I found that the textbook wrote the solution as follows:

    clarification with simple differential equation-codecogseqn-36.gif

    Are these solutions the same? How would I simplify my answer to get the textbook's answer? Just looking to clarify my solution here.
    (Not sure if this belongs in the Calculus forum or the Differential Equations forum. I apologize greatly if I chose the wrong forum to post this question in.)
    Last edited by zachd77; January 4th 2013 at 03:26 PM.
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  2. #2
    Newbie russo's Avatar
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    Re: clarification with simple differential equation

    As far as I know, both answers are right. Since 6 and 2 are constants, c4 and c3 "absorb" them. Then when you find a particular solution, it doesn't matter the way it's written, the constants will adequate to the right result. For example, let's say you find your first constant from your solution and it's 6, then you'll get 6/6 * x^3. If you find the constants using the textbook solution you'll find that the first constant is equal to 1.
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  3. #3
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    Re: clarification with simple differential equation

    Absolutely right both the solutions are right.
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: clarification with simple differential equation

    Another approach would be to observe the characteristic root r=0 is of multiplicity 4, hence the general solution is:

    y(x)=c_1+c_2x+c_3x^2+c_4x^3
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  5. #5
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    Re: clarification with simple differential equation

    russo's answer is correct. So your c_1 is 6 times the book's c_4, your c_2 is 2 times the book's c_3, your c_3 is the book's c_2, and your c_4 is the book's c_1.

    If you had additional information, like maybe initial values or boundary values, you would figure out what all the constants are, and the two general solutions would lead to the same specific solution.

    ibdutt and MarkFL2 are also correct, of course. I didn't see their posts until after I posted mine.

    - Hollywood
    Last edited by hollywood; January 5th 2013 at 07:33 AM.
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