mathematics

Linear differential equation

Linear 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 Linear differential equation rather than just read about it. In short: In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form a 0 ( x ) y + a 1 ( x ) y ′ + a 2 ( x ) y ″ ⋯ + a n ( x ) y ( n ) = b ( x ) {\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a_{n}(x)y^{(n)}=b(x)} where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be line…

Key takeaways

Linear 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 Linear differential equation to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Linear differential equation from memory before moving on to harder problems.

Reference excerpt

In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form

a 0 ( x ) y + a 1 ( x ) y ′ + a 2 ( x ) y ″ ⋯ + a n ( x ) y ( n ) = b ( x ) {\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a_{n}(x)y^{(n)}=b(x)}

where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x. Such an equation is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives.

Types of solution

A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any. The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.

Basic terminology The highest order of derivation that appears in a (linear) differential equation is the order of the equation. The term b(x), which does not depend on the unknown function and its derivatives, is sometimes called the constant term of the equation (by analogy with algebraic equations), even when this term is a non-constant function. If the constant term is the zero function, then the differential equation is said to be homogeneous, as it is a homogeneous polynomial in the unknown function and its derivatives. The equation obtained by replacing, in a linear differential equation, the constant term by the zero function is the associated homogeneous equation. A differential equation has constant coefficients if only constant functions appear as coefficients in the associated homogeneous equation. A solution of a differential equation is a function that satisfies the equation. The solutions of a homogeneous linear differential equation form a vector space. In the ordinary case, this vector space has a finite dimension, equal to the order of the equation. All solutions of a linear differential equation are found by adding to a particular solution any solution of the associated homogeneous equation.

Linear differential operator

A basic differential operator of order i is a mapping that maps any differentiable function to its ith derivative, or, in the case of several variables, to one of its partial derivatives of order i. It is commonly denoted

d i d x i {\displaystyle {\frac {d^{i}}{dx^{i}}}}

in the case of univariate functions, and

∂ i 1 + ⋯ + i n ∂ x 1 i 1 ⋯ ∂ x n i n {\displaystyle {\frac {\partial ^{i_{1}+\cdots +i_{n}}}{\partial x_{1}^{i_{1}}\cdots \partial x_{n}^{i_{n}}}}}

in the case of functions of n variables. The basic differential operators include the derivative of order 0, which is the identity mapping. A linear differential operator (abbreviated, in this article, as linear operator or, simply, operator) is a linear combination of basic differential operators, with differentiable functions as coefficients. In the univariate case, a linear operator has thus the form

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Linear differential equation

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

In research: Linear 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 Linear 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: Linear differential equation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for Linear 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Linear differential equation in 20 minutes

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

Frequently asked questions

What is Linear differential equation in simple terms?

In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form a 0 ( x ) y + a 1 ( x ) y ′ + a 2 ( x ) y ″ ⋯ + a n ( x ) y ( n ) = b ( x ) {\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''\cdots +a…

Why does Linear 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 Linear 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 Linear differential equation.