mathematics

Integrating factor

Integrating factor is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Integrating factor rather than just read about it. In short: In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve non-exact ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a sca…

Key takeaways

Integrating factor belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Integrating factor to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Integrating factor from memory before moving on to harder problems.

Reference excerpt

In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve non-exact ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.

Use An integrating factor is any expression that a differential equation is multiplied by to facilitate integration. For example, the nonlinear second order equation

d 2 y d t 2 = A y 2 / 3 {\displaystyle {\frac {d^{2}y}{dt^{2}}}=Ay^{2/3}}

admits d y d t {\textstyle {\frac {dy}{dt}}} as an integrating factor:

d 2 y d t 2 d y d t = A y 2 / 3 d y d t . {\displaystyle {\frac {d^{2}y}{dt^{2}}}{\frac {dy}{dt}}=Ay^{2/3}{\frac {dy}{dt}}.}

To integrate, note that both sides of the equation may be expressed as derivatives by going backwards with the chain rule:

d d t ( 1 2 ( d y d t ) 2 ) = d d t ( A 3 5 y 5 / 3 ) . {\displaystyle {\frac {d}{dt}}\left({\frac {1}{2}}\left({\frac {dy}{dt}}\right)^{2}\right)={\frac {d}{dt}}\left(A{\frac {3}{5}}y^{5/3}\right).}

Therefore,

( d y d t ) 2 = 6 A 5 y 5 / 3 + C 0 . {\displaystyle \left({\frac {dy}{dt}}\right)^{2}={\frac {6A}{5}}y^{5/3}+C_{0}.}

where C 0 {\displaystyle C_{0}} is a constant. This form may be more useful, depending on application. Performing a separation of variables will give

∫ y ( 0 ) y ( t ) d y 6 A 5 y 5 / 3 + C 0 = t {\displaystyle \int _{y(0)}^{y(t)}{\frac {dy}{\sqrt {{\frac {6A}{5}}y^{5/3}+C_{0}}}}=t}

This is an implicit solution which involves a nonelementary integral. This same method is used to solve the period of a simple pendulum.

Solving first order linear ordinary differential equations Integrating factors are useful for solving ordinary differential equations that can be expressed in the form

y ′ + P ( x ) y = Q ( x ) {\displaystyle y'+P(x)y=Q(x)}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Integrating factor

Start with the simplest possible case. Write down what Integrating factor claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Integrating factor before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Integrating factor ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Integrating factor

In research: Integrating factor appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Integrating factor in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Integrating factor is common in secondary-school and first-year university syllabi. It links to neighbouring topics Ordinary differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for Integrating factor outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Integrating factor in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Integrating factor means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Integrating factor out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Integrating factor in simple terms?

Why does Integrating factor matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Integrating factor?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Integrating factor.