Differential Equations I Essentials
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Differential Equations I Essentials - The Editors of REA
TRANSFORM
CHAPTER 1
BASIC DEFINITIONS AND CLASSIFICATIONS
A differential equation is an equation which involves derivatives of an unknown function.
An ordinary differential equation is an equation which involves only ordinary derivatives.
A partial differential equation is an equation which involves partial derivatives.
The order of a differential equation is the order of the highest derivative that appears in the equation.
The degree of a differential equation is the greatest exponent of the highest ordered derivative in the equation.
A linear ordinary differential equation has the general form
where a0 is not identically zero.
The conditions for a linear equation are:
The dependent variable and its derivatives occur only in the first degree.
No product of the dependent variable and/or any of its derivatives are present.
No transcendental function of the dependent variable and/or its derivatives occur.
The question of existence: Does an equation have a solution or not?
The question of uniqueness: Given an equation with a solution, does it have other solutions? If so, what type of additional conditions must be specified to single out a particular solution?
The general solution (or primitive) of a differential equation is a relation between the variables which involves n arbitrary constants.
The constants in the general solution are called essential if they cannot be replaced by a smaller number of constants.
The primitive is the equation of a family of curves called integral curves. A particular solution is obtained from the primitive by assigning definite values to the arbitrary constants, and is identified as one of the integral curves.
A primitive involving n essential arbitrary constants will give rise to a differential equation of order n with no arbitrary constants. This equation is obtained when you eliminate the n constants by differentiating the primitive n times with respect to the independent