mathematics

Homogeneous differential equation

Homogeneous differential equation 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 Homogeneous differential equation rather than just read about it. In short: A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written f ( x , y ) d y = g ( x , y ) d x , {\displaystyle f(x,y)\,dy=g(x,y)\,dx,} where f and g are homogeneous functions of the same degree of x and y.

Key takeaways

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

Reference excerpt

A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written

f ( x , y ) d y = g ( x , y ) d x , {\displaystyle f(x,y)\,dy=g(x,y)\,dx,}

where f and g are homogeneous functions of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form

d x x = h ( u ) d u , {\displaystyle {\frac {dx}{x}}=h(u)\,du,}

which is easy to solve by integration of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.

History The term homogeneous was first applied to differential equations by Johann Bernoulli in section 9 of his 1726 article De integraionibus aequationum differentialium (On the integration of differential equations).

Homogeneous first-order differential equations

A first-order ordinary differential equation in the form:

M ( x , y ) d x + N ( x , y ) d y = 0 {\displaystyle M(x,y)\,dx+N(x,y)\,dy=0}

is a homogeneous type if both functions M(x, y) and N(x, y) are homogeneous functions of the same degree n. That is, multiplying each variable by a parameter λ, we find

M ( λ x , λ y ) = λ n M ( x , y ) and N ( λ x , λ y ) = λ n N ( x , y ) . {\displaystyle M(\lambda x,\lambda y)=\lambda ^{n}M(x,y)\quad {\text{and}}\quad N(\lambda x,\lambda y)=\lambda ^{n}N(x,y)\,.}

Thus,

M ( λ x , λ y ) N ( λ x , λ y ) = M ( x , y ) N ( x , y ) . {\displaystyle {\frac {M(\lambda x,\lambda y)}{N(\lambda x,\lambda y)}}={\frac {M(x,y)}{N(x,y)}}\,.}

Solution method In the quotient M ( t x , t y ) N ( t x , t y ) = M ( x , y ) N ( x , y ) {\textstyle {\frac {M(tx,ty)}{N(tx,ty)}}={\frac {M(x,y)}{N(x,y)}}} , we can let t = ⁠1/x⁠ to simplify this quotient to a function f of the single variable ⁠y/x⁠:

M ( x , y ) N ( x , y ) = M ( t x , t y ) N ( t x , t y ) = M ( 1 , y / x ) N ( 1 , y / x ) = f ( y / x ) . {\displaystyle {\frac {M(x,y)}{N(x,y)}}={\frac {M(tx,ty)}{N(tx,ty)}}={\frac {M(1,y/x)}{N(1,y/x)}}=f(y/x)\,.}

That is

d y d x = − f ( y / x ) . {\displaystyle {\frac {dy}{dx}}=-f(y/x).}

Introduce the change of variables y = ux; differentiate using the product rule:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Homogeneous differential equation

Start with the simplest possible case. Write down what Homogeneous differential equation 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 Homogeneous differential equation 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 Homogeneous differential equation 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 Homogeneous differential equation

In research: Homogeneous differential equation 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 Homogeneous differential equation 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: Homogeneous differential equation 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 Homogeneous differential equation 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 Homogeneous differential equation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Homogeneous differential equation 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 Homogeneous differential equation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Homogeneous differential equation in simple terms?

A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written f ( x , y ) d y = g ( x , y ) d x , {\displaystyle f(x,y)\,dy=g(x,y)\,dx,} where f and g are homogeneous functions of the same degree of x and y.

Why does Homogeneous differential equation 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 Homogeneous differential equation?

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 Homogeneous differential equation.