Differential equation integrating factor conceptual question.

In summary, the question asks why the absolute value is omitted in the integration when using the integrating factor 1/x. The answer is that since 1/x is not continuous at x=0, it cannot be "continued" across x=0 and we must choose to restrict x to be positive or negative. This results in the same solution regardless of which integrating factor is chosen.
  • #1
td21
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Homework Statement


Sorry asking similar quesion again about absolute value. You can read the attachment.
u(x) is the integrating factor. Why absolute value is omitted in the integration? and why the integrating factor is not "1/|x|", with the absolute sign

Homework Equations





The Attempt at a Solution


i thought we must use the abs sign and the solution is wrong?
 

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  • #2
The answer is exactly the same as to the question you asked in
https://www.physicsforums.com/showthread.php?t=474430

Since 1/x is not continuous at x= 0, it cannot be "continued" across x= 0. We must choose to restrict x to be positive or negative. If we choose to restrict x to be positive, we have 1/x. If we choose to restrict x to be negative, we have -1/x.

Since you use the integrating factor by multiplying the entire equation by it, you will get the same solution either way:

Taking the integrating factor to be 1/x, we get
[tex]\frac{1}{x}\frac{dy}{dx}- \frac{y}{x^2}= \frac{d}{dx}\left(\frac{1}{x}y\right)= 2+ \frac{1}{x}[/tex]

Taking the integrating factor to be -1/x, we have
[tex]\frac{-1}{x}\frac{dy}{dx}+ \frac{y}{x^2}= -\frac{d}{dx}\left(\frac{1}{x}y\right)= -2- \frac{1}{x}[/tex]
But multiplying both sides of that equation by -1 gives the same as the first equation.
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates one or more functions and their derivatives. It describes the rate of change of a dependent variable with respect to one or more independent variables.

2. What is an integrating factor?

An integrating factor is a function that is multiplied to both sides of a differential equation to convert it into an exact differential equation. This allows for easier integration and solution of the differential equation.

3. How do you find the integrating factor for a differential equation?

To find the integrating factor for a differential equation, you must first identify the form of the equation and then use a specific method to determine the integrating factor. For example, for a linear differential equation, the integrating factor is the exponential of the integral of the coefficient of the dependent variable.

4. What is the purpose of using an integrating factor?

The purpose of using an integrating factor is to convert a non-exact differential equation into an exact one, making it easier to solve. It essentially helps to simplify the integration process and find a general solution to the differential equation.

5. Can all differential equations be solved using an integrating factor?

No, not all differential equations can be solved using an integrating factor. This method is only applicable to certain types of equations, such as linear, first-order equations. Other methods, such as separation of variables or substitution, may be needed to solve different types of differential equations.

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