# Stumped by Integral: Solving a Differential Equation

• brendan_foo
In summary, Daniel came up with an integral that has stumped him. He used an integrating factor and got the same result as brenden. He isn't sure what the equivalent is, but is fairly proficient up to Calc III. He is looking for a suitable text in which he can do some self study and tutor himself as best possible.
brendan_foo
Hi guys,

In an attempt to solve the following differential equation, I have come up with an integral that has stumped me.

The differential equation is as follows:

$$\frac{dy}{dx} + \frac{y}{x^2} = 2x$$

Using an integrating factor, I end up with the following:

$$y \cdot e^{-\frac{1}{x}} = 2\int xe^{-\frac{1}{x}} dx$$

I cannot solve that right hand integral, I have tried using parts and substitution and I can't really yield anything meaningful... Is it possible to evaluate this integral using basic calculus methods? Or is something else required?

Thanks!

Add a constant to this result.

Daniel.

#### Attachments

• Integrate.gif
1.2 KB · Views: 457
I tried using the integrator from Wolfram..but I wanted to see if there was a neater result. This was actually a question on an engineering paper. Have I done steps preceeding the integral evaluation correct, with the integrating factor?

Everything is okay.Here's what Maple says

$$\frac{dy}{dx}+\frac{y}{x^2}=2x$$

, Exact solution is :

$$y\left( x\right) =x^2-x+e^{\frac{1}{x}}\mbox{Ei}\left( 1,\frac{1}{x}\right) +Ce^{\frac{1}{x}}$$

Daniel.

I get the same thing as you brenden.

Also, I wouldn't trust that integrator too much. It has hapened to me twice that he provided a wrong answer.

Ok cheers fellas, must've been a type-o in the paper.

Much appreciated.

Oh by the way, I'm not sure what the equivalent is, but I am fairly proficient up to Calc III, and I want to begin pursuing some advance calculus. I am looking for a suitable text in which I can do some self study and tutor myself as best possible. Can anyone recommened a thorough and lucid text for self-study?

I am looking for something more precise, as opposed to just a list of rules and how to implement them.

Thanks guys, much appreciated
Peace

Last edited:

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It describes the relationship between a function and its rate of change.

## 2. What is an integral?

An integral is a mathematical operation that represents the area under a curve on a graph. It is the inverse operation of differentiation and is used to solve differential equations.

## 3. How do you solve a differential equation using integration?

To solve a differential equation using integration, you first need to identify the type of differential equation and then use integration techniques to find a general solution. This solution can then be further manipulated to find specific solutions for given initial conditions.

## 4. What is the difference between an ordinary differential equation and a partial differential equation?

An ordinary differential equation involves one independent variable and its derivatives, while a partial differential equation involves multiple independent variables and their derivatives. Ordinary differential equations are often used to model physical systems, while partial differential equations are used to describe phenomena that vary in space and time.

## 5. What are some common applications of solving differential equations?

Solving differential equations is used in a variety of fields, including physics, engineering, economics, and biology. It can be used to model and predict the behavior of systems in these fields, such as population growth, heat transfer, and electrical circuits.

• Calculus
Replies
2
Views
2K
• Calculus
Replies
2
Views
1K
• Calculus
Replies
4
Views
1K
• Calculus
Replies
19
Views
3K
• Calculus
Replies
1
Views
2K
• Calculus
Replies
49
Views
3K
• Calculus
Replies
6
Views
2K
• Calculus
Replies
4
Views
2K
• Calculus
Replies
9
Views
2K
• Calculus
Replies
3
Views
1K