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Math Help - Differential Equation

  1. #1
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    Differential Equation

    I was having trouble with this question as I'm not sure how to split it into dy and dx in order to integrate.

    x*dy/dx+y = x given y(-1)=2
    .
    y= ?


    Thanks, Daddy_Long_Legs
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  2. #2
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    You solve it using integrating factor
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  3. #3
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    songoku,

    I'm not sure what you mean by that. I need to see what you do for the first step thanks.
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  4. #4
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     x \frac{dy}{dx} + y = x

    \frac{dy}{dx} + \frac{y}{x} = 1

    Have you learnt how to solve linear form differential equation using integrating factor?
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  5. #5
    Senior Member Danneedshelp's Avatar
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    y'+\frac{y}{x}=1

    find your integrating factor...

    \mu{(x)}=e^{\int{\frac{1}{x}}dx}\Rightarrow\mu{(x)  }=x

    Multiply both sides by your integrating factor and integrate...

    \int{(yx)'}=\int{x}

    yx=\frac{x^{2}}{2}+c

    Solve for C

    2(-1)=\frac{(-1)^{2}}{2}+C

    C=-\frac{5}{2}

    Now, solve for y ( f(x)) and you are done.

    f(x)=\frac{x}{2}-\frac{5}{2x}

    Seems like you have never been introduced to this method, so check out this wiki page: Integrating factor - Wikipedia, the free encyclopedia

    It's pretty straight forward, just takes some practice.
    Last edited by Danneedshelp; August 4th 2009 at 11:28 AM.
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  6. #6
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    Thanks songoku and Danneedshelp,

    I haven't come across any questions were x and y weren't products so this is new to me. I'll have a look at the link Danneedshelp. Thanks.
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  7. #7
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    Danneedshelp,

    I've read this link and I'm still stuck with this question, I have followed your working but I'm afraid I can't see how you got the answer. I was wondering if you or another member could show me again as this question is coming up repeatedly and I'm still clueless on to do an intergrading factor.

    Thanks.
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  8. #8
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    Hi Daddy_Long_Legs

    Maybe this will be helpful^^
    http://www.mathhelpforum.com/math-he...-tutorial.html
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  9. #9
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    Danneedshelp,

    I found y=x^2/2+3/2, to be my final answer as I divided by x and then found the constant, I'm not sure which way this should happen?

    Thanks.
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  10. #10
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    Hi Daddy_Long_Legs

    There's typo in Danneedshelp post, the solution should be <br />
f(x)=\frac{x}{2}-\frac{5}{2x}<br />

    Can you please post your work so I can take a look at it ? ^^
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  11. #11
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    Songoku,

    I see where I went wrong with my working, I tried to cross multiply when I got to y=x/2+c/x. If I solve c from this step then I can the same answer you mentioned. (I attached my working below)

    Thanks.
    Attached Thumbnails Attached Thumbnails Differential Equation-differential.jpg  
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  12. #12
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  13. #13
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by songoku View Post
    Hi Daddy_Long_Legs

    There's typo in Danneedshelp post, the solution should be <br />
f(x)=\frac{x}{2}-\frac{5}{2x}<br />

    Can you please post your work so I can take a look at it ? ^^
    My bad!

    Sorry about that Daddy_Long_legs!!!
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  14. #14
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    songoku and Danneedshelp,

    I have come across this question a few times in my reviews and I have entered this answer several times and I keep getting the question wrong and I'm not sure why? The question mentions parenthesizing the denominator carefully but I have two dominators which leads me to believe that the answer I have been entering is incorrect or that it needs rearranging of some kind.

    Thanks.
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  15. #15
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    Quote Originally Posted by Daddy_Long_Legs View Post
    songoku and Danneedshelp,

    I have come across this question a few times in my reviews and I have entered this answer several times and I keep getting the question wrong and I'm not sure why? The question mentions parenthesizing the denominator carefully but I have two dominators which leads me to believe that the answer I have been entering is incorrect or that it needs rearranging of some kind.

    Thanks.
    A perfectly good answer, already given, is y = \frac{x}{2} - \frac{5}{2x}.

    I'll bet that whoever gave you this question wants the answer written as y = \frac{x^2 - 5}{2x}, which is completely pointless since the point of the question, I would have thought, is to test whether you can solve a DE, not whether you can read the (narrow and small) mind of someone.

    (A likely scenario is that the person who gave you this question has copied it from somewhere and is slavishly using the given answer rather than exercising some common sense and/or intelligence).
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