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How to get started on this?

  1. Apr 14, 2005 #1
    Q. If dy/dx = e^x / x and y(1) = 2; find an approximate value for y(3). Use a technique from calculus or technology to help you solve the problem. It is impossible to find an antiderivative.

    My thoughts / ideas:

    I thought this was a separable equation, and could separate the x and y variables and then may be just integrate both sides.

    But I don't think this is possible, Since the question clearly says "It is impossible to find an antiderivative".

    Any ideas.
  2. jcsd
  3. Apr 14, 2005 #2
    Linear approximations? Eulers method?
  4. Apr 14, 2005 #3
    Yeah, I think you are right, Euler's Method would work definetly.

    How about using calculus. any ideas.

    I can make a 'spreadsheet' in Excel that can calculate the differential at the specified point

    using Euler's Method and Euler's Improved method.

    But any ideas on how to actually use calculus.
  5. Apr 14, 2005 #4
    Euler's method and linear approximations are calculus methods.
  6. Apr 14, 2005 #5
    I used the fact that delta(y) is roughly equal to delta(x) times dy/dx. Then you come up with y(3)-y(1)=(e^1/1)(3-1). I think this gives y(3)= 2e+2. Could someone verify that this is the correct approximation?

    Thanks, Joe
  7. Apr 14, 2005 #6
    Ok, Let us give up technology for a moment ,and actually think , how to solve this problem analytically using calculus.

    I know we could use Euler's Method or Linear Approximation, but how do we apply them analytically .. How to get started?

    Please help!
  8. Apr 14, 2005 #7
    1)I think that you can make fast work on this question by using McClaurin's expansion for e^x, then divide it later by x to find dy/dx (in a summation notation for easy integration later)
    2) For the second part, since the initial value is given we can use the fundamental theorem of calculus to find a short cut to the general form of the solution.
    (i.e) [tex]y=\int_{1}^{x} f(t)dt +2 [/tex]

    f(t) here is simply the series expansion for [tex]e^x/x[/tex]
  9. Apr 14, 2005 #8


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    I resent that and claim it's incorrect,because

    [tex] \int \frac{e^{x}}{x} \ dx =\mbox{Ei}\left(x\right) +C [/tex]

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