How to Approximate Solutions for an Impossible Antiderivative?

  • Thread starter Naeem
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In summary, the problem asks to find an approximate value for y(3) given the differential equation dy/dx = e^x / x and initial value y(1) = 2. It is stated that it is impossible to find an antiderivative for this equation. Techniques from calculus or technology can be used to solve the problem. Ideas such as Euler's method, linear approximations, and using Excel were discussed. Ultimately, the solution was found using McClaurin's expansion for e^x and applying the fundamental theorem of calculus. This resulted in the general form of the solution, y = ∫1xdte^t + 2.
  • #1
Naeem
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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.
 
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  • #2
Linear approximations? Eulers method?
 
  • #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.
 
  • #4
Euler's method and linear approximations are calculus methods.
 
  • #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
 
  • #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!
 
  • #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]
 
  • #8
Naeem said:
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.


I resent that and claim it's incorrect,because

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


Daniel.
 

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