Lecture

Linear differential equations

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Description

This lecture covers linear differential equations of order n, defining them as equations of the form y^(n)(x) + g_n-1(x) y^(n-1)(x) + ... + g_1(x) y'(x) + g_0(x) y(x) = f(x). It explores homogeneous and inhomogeneous equations, solutions, linear independence, and the existence and uniqueness theorem. The lecture also presents examples showcasing linearly independent functions like cos(x) and sin(x). Theorems are introduced to explain the existence of n linearly independent solutions for an nth-order homogeneous equation, and the general solution is derived from n linearly independent solutions. The proofs are briefly outlined.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Ontological neighbourhood

Mathematics

Analysis: Differential anaysis

Algebra: Linear algebra

Related concepts (32)

Homogeneous differential equation

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 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 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.

Linear differential equation

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form 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).

Linear independence

In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.

Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .

System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple since it makes all three equations valid. The word "system" indicates that the equations should be considered collectively, rather than individually.

Related lectures (64)

Advanced Analysis II: Second Order Linear Differential Equations

Covers second order linear differential equations, focusing on solution construction and the concept of linear independence among solutions.

General Solution of Homogeneous Second Order Linear Differential Equations

Covers the general solution of homogeneous second-order linear differential equations with constant coefficients and the concept of linear independence of solutions.

Linear Equations: Least Squares Solution MOOC: Linear Algebra (Part 3)

Explains how to solve linear equations using the least squares method to minimize errors in the system.

General Solutions of Differential Equations

Explores general solutions of differential equations, emphasizing homogeneous and inhomogeneous equations and the process of finding solutions.

Fourier Transform: LTI Systems