Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Linear Differential Equations of Higher Order

In course

MATH-105(b): Advanced analysis II

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables. ...

Description

This lecture covers the revision of linear differential equations with constant coefficients of higher order, focusing on the resolution of the associated homogeneous equation and the interpretation of the solutions geometrically. It also explores the general solution of the problem, emphasizing the qualitative solutions and the method of variation of constants.

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

Related concepts (32)

Exact solutions in general relativity

In general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter. Mathematically, finding an exact solution means finding a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field.

Fluid solution

In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid. In astrophysics, fluid solutions are often employed as stellar models. (It might help to think of a perfect gas as a special case of a perfect fluid.) In cosmology, fluid solutions are often used as cosmological models.

Vacuum solution (general relativity)

In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non-gravitational fields are present. These are distinct from the electrovacuum solutions, which take into account the electromagnetic field in addition to the gravitational field.

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).

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.

Related lectures (40)

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.

Advanced Analysis II: Variation of Constants Method

Covers the variation of constants method for solving first-order linear differential equations, detailing its steps and implications for general and particular solutions.

Method of Undetermined Coefficients

Explores the method of undetermined coefficients for solving non-homogeneous linear differential equations with constant coefficients.

Linear Differential Equations

Covers second-order linear differential equations and their solutions, including the method of variation of constants.

Differential Equations: Coefficients and Methods