integrating factor

USAGE SUMMARY

The phrase "integrating factor" is correct and usable in written English.
It is typically used in mathematics, particularly in the context of solving differential equations, to refer to a function that simplifies the equation. Example: "To solve the first-order linear differential equation, we need to find the integrating factor."

✓ Grammatically correct

Science

Academia

Formal & Business

Human-verified examples from authoritative sources

Exact Expressions

33 human-written examples

Moreover, the integrability of a class of systems (quasidegenerate infinity systems) which generalize the so-called degenerate infinity vector fields is considered, and a method how to obtain an inverse integrating factor of the systems from the first integrals of the corresponding quasihomogeneous systems is shown.

Advances in Difference Equations

where the factor 1 P V Open image in new window would represent the integrating factor of the infinitesimal work δW HP that makes the integration function an exact differential function.

International Journal of Energy and Environmental Engineering

We use the method of integrating factor of Wang et al. [18].

Journal of Inequalities and Applications

By solving the systems of ordinary differential equations which are established by using the vector fields of the quasihomogeneous systems, one can obtain an inverse integrating factor of the systems.

Advances in Difference Equations

In particular, Wang et al. [18] used the method of integrating factor to prove the Hyers-Ulam stability of the nonhomogeneous linear differential equations of the form p(t y'(t) - q(t y(t) - r(t) = 0. (1).

Journal of Inequalities and Applications

Fortunately, Yirong Liu and Jibin Li [21] found that there always exists a formal inverse integrating factor for three-order nilpotent critical points in 2009, and they gave a new definition of the focal values under the generalized triangle polar coordinates and the method of commuting Lyapunov constants using the inverse integral factors for the three-order nilpotent critical point.

Advances in Difference Equations

Human-verified similar examples from authoritative sources

Similar Expressions

26 human-written examples

The proposed methodology integrates factor productivity and relative price analysis with input output model by using V-RAS method.

Telematics and Informatics

We refer to this model as Bayesian switching factor analysis (BSFA) as it integrates factor analysis into a generative HMM in a unified Bayesian framework.

NeuroImage

We now need to integrate factors comprehensively to evaluate the high-temperature performance of asphalt and its mixtures.  .

Journal of Modern Transportation

In [98], authors studied game dynamics and learning schemes and proposed a novel learning scheme called cost-to-learn which integrates factors such as switching cost and delay as well as user behavior.

EURASIP Journal on Wireless Communications and Networking

For the first time, to our knowledge, the obtained HF classification integrated factors that have a large impact on the prevalence of symptoms and signs suggestive of HF.

BMJ Open

Expert writing Tips

Best practice

When using the term "integrating factor", ensure the context clearly relates to differential equations or mathematical analysis where such factors are applied to simplify and solve equations. Provide sufficient context for non-expert readers to understand its role.

Common error

Avoid using "integrating factor" in contexts outside of mathematics or physics where it has no established meaning. Overgeneralizing its use can lead to confusion. Instead, opt for more general terms like "key component" or "essential element" in non-technical contexts.

Linguistic Context

The phrase "integrating factor" functions as a noun phrase, specifically in mathematical contexts. It identifies a function used to simplify differential equations. As Ludwig AI indicates, it is primarily found in technical literature.

Expression frequency: Common

Frequent in

Science

75%

Academia

15%

Formal & Business

5%

Less common in

News & Media

2%

Encyclopedias

1%

Wiki

2%

Ludwig's WRAP-UP

The phrase "integrating factor" is a mathematical term used to simplify differential equations. As Ludwig AI confirms, it is grammatically correct and primarily used in formal and scientific contexts. While alternatives like "multiplicative factor" exist, "integrating factor" is the standard term in mathematical literature. The frequency of use is common, and it is essential to ensure its correct application within relevant mathematical discussions. Misunderstanding its applicability outside of these contexts should be avoided. Its primary function is to denote and describe a tool used in mathematical methodologies.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

factor of integration

A slightly reordered version of the original phrase, maintaining the same meaning.

integrating multiplier

Synonymous term, highlighting the multiplicative nature of the factor.

inverse integrating factor

Refers to the reciprocal of the standard integrating factor, used in specific mathematical techniques.

multiplicative factor

This term emphasizes the mathematical operation performed by the "integrating factor", focusing on multiplication.

adjustment factor

Highlights the role of the "integrating factor" in modifying an equation to make it solvable.

normalizing factor

Implies that the "integrating factor" is used to bring a function or equation into a standard or normalized form.

solvability factor

This term emphasizes the function of the "integrating factor" in enabling the solution of a differential equation.

auxiliary function for integration

More descriptive, specifying the role of the function in the integration process.

linearizing factor

If the integrating factor transforms a non-linear equation to a linear one, this phrase is applicable.

function for simplifying integration

Highlights the purpose of the function, which is to make the integration process easier.

FAQs

What is the purpose of using an "integrating factor" in mathematics?

An "integrating factor" is used to transform a non-exact differential equation into an exact one, making it easier to solve.

How do I find an "integrating factor" for a given differential equation?

The method for finding an "integrating factor" depends on the form of the differential equation. For linear first-order equations, a standard formula exists; for others, it may require more complex techniques.

Are there alternatives to using an "integrating factor" to solve differential equations?

Yes, alternative methods include using Laplace transforms, numerical methods, or seeking series solutions, depending on the nature of the equation.

What is the difference between an "integrating factor" and a "normalizing factor"?

While both are multiplicative factors used in mathematical contexts, an "integrating factor" specifically aids in solving differential equations, while a "normalizing factor" typically scales a function to have a specific property, such as unit area or probability.