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First Order OEDs


Differential Equations!Unit I: First Order Differential Equations

Contents

Introduction

Unit I: 1st Order ODEs

A differential equation is an equation containing a derivative.

Some examples are {$ {d \over dx } y = {x \over y} $}, {$ y^\prime = y - x^2 $}, and {$ \color{blue}{ y^\prime = x^2 - y} $}.

Most of this course pertains to ODE, ordinary differential equations, which are differential equations with only one independent variable. The first unit deals with ODEs of the first order which have only the first derivative.

Most first order differential equations look like (or can be made to look like):

{$$ y'= f(x,y) \tag{Standard Form} $$}

Some DE's are not easily fit into standard form.

This is not the standard form of DEs considered as linear equations. This form must not be used in an attempt to solve a DE as a linear equation.

There are three basic approaches to solving differential equations:

Analytic
This approach uses calculus and algebra. It provides an accurate and rigorous solution when it is possible to use it. Unfortunately, many differential equations cannot be solved by this method.
Geometric
This approach uses human or computer generated graphs to plot some aspects of the equation which can be known. Approximate solutions are then fitted into the graph.
Numerical
This used a series of approximations to solve for an approximate solution. It is tedious and generally done by a computer. Approximations generally have a systemic error. In some cases, this approach can blow up completely, as may happen when there is an unexpected discontinuity.

<< Differential Equations Mattuck | Trail MIT 18.03SC Differential Equations | Conventions >>

Sources:

Edwards, C., and D. Penney. Elementary Differential Equations with Boundary Value Problems. 6th ed. Upper Saddle River, NJ: Prentice Hall, 2003 [EP]

Recommended:

[EP]: 1.1 and 1.4

Categories:

Tags: Xxx First Order Differential Equations , Xxx Ordinary Differential Equations

Includes:


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