Integrating Factor and absolute value

• awelex
In summary: So we can just drop the absolute value sign and say that an integrating factor is any function, \mu(x), such that \mu'(x)= \mu(x)f(x)!In summary, the integrating factor for a linear first order equation is any function that satisfies \mu'(x)= \mu(x)f(x). The absolute value symbol is often dropped when the factor has the form exp(ln(abs(x))), as the positive or negative value does not affect the overall solution.
awelex
Hi,

I have a general question regarding the integrating factor of first-oder linear DEs. All textbooks that I've seen (which aren't too many) simply drop the absolute symbol when the factor has the form exp(ln(abs(x))). This would evaluate to abs(x), yet the books use simply x. Why is that valid?

Thanks,

Alex

awelex said:
Hi,

I have a general question regarding the integrating factor of first-oder linear DEs. All textbooks that I've seen (which aren't too many) simply drop the absolute symbol when the factor has the form exp(ln(abs(x))). This would evaluate to abs(x), yet the books use simply x. Why is that valid?

Thanks,

Alex

I've been wondering too.

Last edited:
An "integrating factor" for a linear first order equation, dy/dx+ f(x)y= g(x), is a function, $\mu(x)$ such that if we multiply the entire equation by it, $\mu(x)(y'+ f(x)y)= \mu(x)y'+ f(x)\mu(x)y= \mu(x)g(x)$, the left side becomes an "exact derivative":
$$\frac{d\mu(x)y}{dx}$$

If that is true then, by the product rule,
$$\frac{d\mu(x)y}{dx}= \mu(x)y'+ \mu' y= \mu(x)y'+ f(x)\mu(x)y$$
which leads immediately to $\mu'(x)= x\mu(x)$, a simple separable equation.
$$\frac{d\mu}{\mu}= f(x)dx$$
so that $ln(|\mu|)= \int f(x)dx$ and so
$$|\mu|= e^{\int f(x)dx}$$
Of course, that means that either $\mu= e^{\int f(x)dx}$ or $\mu(x)= -e^{\int f(x)dx}$

But since we are multiplying both sides of the equation by $\mu(x)$ it really doesn't matter whether it is positive or negative!

Recoil

What is an integrating factor?

An integrating factor is a function that is used in the process of solving a differential equation. It is multiplied by the entire equation in order to make it easier to solve.

Why do we need to use an integrating factor?

Integrating factors are used to simplify the process of solving differential equations. They allow us to transform a more complex equation into a simpler one that is easier to solve using standard methods.

What is the role of absolute value in integrating factor?

Absolute value is used in integrating factor when solving differential equations that involve negative values. It is used to ensure that the solution is valid for both positive and negative values of the variable.

Can integrating factor be used for all types of differential equations?

Integrating factor can be used for certain types of differential equations, specifically those that can be written in the form of a first-order linear differential equation. It cannot be used for non-linear differential equations.

How do you determine the integrating factor for a given differential equation?

The integrating factor can be determined by finding the integrating factor equation, which is a function of the independent variable. This equation can be found by dividing the coefficients of the differential equation by the coefficient of the highest order derivative term.

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