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Homework Help: Linear DE

  1. Jan 23, 2007 #1
    1. The problem statement, all variables and given/known data

    Solve xy' + 2y = sqrt(1 + x^2)

    2. Relevant equations

    I got an integrating factor of [itex]x^2[/itex]

    So [tex]y = 1/x^2 \int{x\sqrt{1 + x^2}}[/tex]

    3. The attempt at a solution

    and I get [tex]y =\frac{(1 + x^2)^{3/2}}{3x^2}[/tex] for C = 0

    But this wrong when I substitute it in ... what'd I do wrong?
    Last edited: Jan 23, 2007
  2. jcsd
  3. Jan 23, 2007 #2
    How would I solve this in Maple? When I do and try it's answer it seems wrong as well... it tells me [itex]2y = \sqrt(1+x^2)[/tex]
  4. Jan 23, 2007 #3


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    It is wrong because you have multiplied the sqruare root expression with x^2, rather than with just x:

    And so on..
  5. Jan 23, 2007 #4
    sorry, I that was a typo... with [itex]x\sqrt{1 + x^2}[/itex] you get the y as above. (at least I do... on my calculator too) but when I substitute for y into the LHS it doesn't equal [itex]\sqrt{1+x^2}[/itex]
  6. Jan 23, 2007 #5

    D H

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    I just used the Maple packaged with Matlab.

    >> syms x y C1

    >> ysoln = dsolve('x*Dy+2*y=sqrt(1+x^2)','x')
    ysoln = (1/3*(1+x^2)^(3/2)+C1)/x^2

    >> x*diff(ysoln,x)+2*ysoln-sqrt(1+x^2)
    ans = A big awful mess

    >> pretty(simple(ans))

    In other words, the answer you got in the original post is correct.

    Moral: Sometimes Maple (or Mathematica) create a big long mess. Simplification cleans it up, sometimes (as it did in this case).

    Usually when I use Maple I am left with a big long mess that Maple can't simplify. I use Maple to check my hand work for stupid errors.
    >>mysoln = some simple equation;
    >>maplesoln = some big long mess;
    >>zerocheck = simple(mysoln-maplesoln)
    Maple doesn't make the stupid mistakes I am quite prone to make. On the other hand, Maple is quite stupid about simplifying things.
    Last edited: Jan 23, 2007
  7. Jan 23, 2007 #6
    I agree with DH that 'dsolve' is a good tool to use to solve these diffs on maple.
    There are also a lot of great tools in maple to simplify answers. If all your getting on maple are big nasty answers, your not using it right.
    read up on 'simplify' ,'factor', and 'collect', those are the 3 most common tools i use
  8. Jan 23, 2007 #7
    Thanks... for the tips too!
  9. Jan 23, 2007 #8

    D H

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    Don't be insulting. Those tools only go so far. The bottom line is that simplification is not particularly well-defined and is definitely not easy (it is an NP-hard problem). I have run across many problems that neither Maple nor Mathematica can simplify as well as a human can. I have seen many, many more examples in autogenerated code (no human would write a multi-page equation).
  10. Jan 24, 2007 #9
    Sorry, I didnt mean to sound insulting.
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