# How to Approximate Solutions for an Impossible Antiderivative?

• Naeem
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.
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.

Linear approximations? Eulers method?

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.

Euler's method and linear approximations are calculus methods.

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

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?

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 $$e^x/x$$

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

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

Daniel.

## 1. How do I choose a research topic?

Choosing a research topic can be a daunting task, but it's important to choose a topic that you are passionate about and that aligns with your research interests. Consider your previous coursework, current trends in your field, and potential gaps in existing research. It's also helpful to discuss your ideas with colleagues and mentors to get their feedback.

## 2. What is the first step in conducting research?

The first step in conducting research is to clearly define your research question or hypothesis. This will guide your entire research process and help you stay focused on your goals. It's important to make sure your research question is specific, measurable, and relevant to your field.

## 3. How do I conduct a literature review?

A literature review involves researching and analyzing existing literature on your chosen topic. Start by searching for relevant articles, books, and other sources using databases and search engines. As you read, take notes on key findings and ideas. Organize your notes and synthesize the information to identify any gaps or areas for further research.

## 4. What is the best way to collect data?

The best way to collect data will vary depending on your research question and methodology. Common methods include surveys, experiments, interviews, and observations. It's important to carefully plan and design your data collection methods to ensure you are gathering accurate and relevant data.

## 5. How do I analyze my data?

Data analysis involves using statistical and analytical methods to interpret your data and answer your research question. This can include organizing and summarizing your data, identifying patterns and trends, and drawing conclusions. It's important to choose the appropriate analysis techniques for your data and consult with a statistician if needed.

• Calculus
Replies
15
Views
2K
• Calculus
Replies
8
Views
515
• Introductory Physics Homework Help
Replies
1
Views
802
• Calculus
Replies
9
Views
3K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
835
• Calculus and Beyond Homework Help
Replies
14
Views
557
• Differential Equations
Replies
7
Views
2K
• Introductory Physics Homework Help
Replies
3
Views
831
• Calculus
Replies
8
Views
490