Lecture

Ordinary Differential Equations: Separable Variables

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 ordinary differential equations (EDOs) of the form F(x, y, y(1),..., y(n)) = 0, where F is a continuous function defined on an open set in Rn+2. It explains the concept of separable variables in EDOs, where y: R→R is an n times continuously differentiable function. The lecture introduces the notion of a Cauchy problem, providing an EDO and an initial condition. It delves into the definition of an ordinary differential equation with separable variables (EDOv.s.), which involves equations of the form y' = f(x)g(y). Theorems are presented to establish conditions for the existence and uniqueness of solutions in separable variable EDOs.

Official source

Related concepts (28)

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), .

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).

Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

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.

Separation of variables

In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. A differential equation for the unknown will be separable if it can be written in the form where and are given functions. This is perhaps more transparent when written using as: So now as long as h(y) ≠ 0, we can rearrange terms to obtain: where the two variables x and y have been separated.

Related lectures (64)

Advanced Analysis II: Differential Equations and Timers

Discusses advanced analysis concepts, focusing on differential equations and timers in microcontrollers.

Canonical Transformations: Existence and Equations

Explores canonical transformations, focusing on existence, equations, simplicity, and differential equations' theory.

Differential Equations: Solution Methods MOOC: Electrical engineering II

Explores the resolution of differential equations and the properties of exponential functions in electrical engineering.

Differential Equations: Initial Conditions and Solutions

Covers the solution of differential equations with initial conditions and limit analysis.

Material Point Model: Basics