How to Approximate Solutions for an Impossible Antiderivative?

[Naeem]

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.

 

[whozum]

Linear approximations? Eulers method?

 

[Naeem]

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.

 

[whozum]

Euler's method and linear approximations are calculus methods.

 

[josephcollins]

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

 

[Naeem]

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!

 

[estalniath]

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) $$y=\int_{1}^{x} f(t)dt +2$$

f(t) here is simply the series expansion for $$e^x/x$$

 

[dextercioby]

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

$$\int \frac{e^{x}}{x} \ dx =\mbox{Ei}\left(x\right) +C$$


Daniel.