Ordinary differential equations

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1 Definition of a differential equation
2 Order of a differential equation
3 Arbitrary constants
4 Solution of a differential equation
5 Differential equation of a family of curves

Definition of a differential equation

A differential equation is an equation involving derivatives or differentials.

Equations involving only one independent variable are called ordinary differential equations. Equations with two or more independent variables are called partial differential equation.

Order of a differential equation

An equation having a derivative of nth order but no higher is called an nth order differential equation.

Arbitrary constants

An arbitrary constant, often denoted by a letter at the beginning of the alphabet such as A, B, C, c₁, c₂, etc., may assume values independently of the variables involved. For example in $y = x^2 + c_1x + c_2$ , c₁ and c₂ are arbitrary constants.

The relation of $y = Ae^{-4x + B}$ which can be written $y = Ae^Be^{-4x} = Ce^{-4x}$ actually involves only one arbitrary constant. It’s always assumed that the minimum number of constants is present, i.e. the arbitrary constants are essential.

Solution of a differential equation

A solution of a differential equation is a relation between the variables which is free of derivatives and which satisfies the differential equation identically. $y = x^2 + c_1x + c_2$ is a solution of $y'' = 2$ since by substitution the identity 2 = 2.

A general solution of an nth order differential equation is one involving n (essential) arbitrary constants. Since $y = x^2 + c_1x + c_2$ has two arbitrary constants and satisfies the second order differential equation $y'' = 2$ , it is a general solution of $y'' = 2$ .

A particular solution is a solution obtained from the general solution by assigning specific values to the arbitrary constants. $y = x^2 - 3x + 2$ is a particular solution of $y'' = 2$ and is obtained from the general solution $y = x^2 + c_1x + c_2$ by putting c₁ = -3 and c₂ = 2.

A singular solution is a solution which cannot be obtained from the general solution by specifying values of the arbitrary constants. The general solution of $y = xy' - y'^2$ is $y = cx - c^2$ . However, as seen by substitution another solution is $y = x^2/4$ which cannot be obtained from the general solution for any constant c. This second solution is a singular solution.

Differential equation of a family of curves

A general solution of an nth order differential equation has n arbitrary constants (or parameters) and represents geometrically an n parameter family of curves. Conversely a relation with n arbitrary constants (sometimes called a primitive) has associated with it a differential equation of order n (of which it is a general solution) called the differential equation of the family. This differential equation is obtained by differentiating the primitive n times and then eliminating the n arbitrary constants among the n + 1 resulting equations.