\n
Definition of a differential equation<\/h2>\n
A differential equation<\/span> is an equation involving derivatives or differentials. <\/p>\n
Equations involving only one independent variable are called ordinary differential equations<\/span>. Equations with two or more independent variables are called partial differential equation<\/span>. <\/p>\n
Order of a differential equation<\/h2>\n
An equation having a derivative of n<\/span>th order but no higher is called an n<\/span>th order differential equation<\/span>. <\/p>\n
Arbitrary constants<\/h2>\n
An arbitrary constant, often denoted by a letter at the beginning of the alphabet such as A, B, C, c_{1<\/sub>, c_{2<\/sub><\/span>, etc., may assume values independently of the variables involved. For example in , c_{1<\/sub><\/span> and c_{2<\/sub><\/span> are arbitrary constants. <\/p>\n}}}}
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<\/span>. <\/p>\n
Solution of a differential equation<\/h2>\n
A solution<\/span> 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. <\/p>\n
A general solution<\/span> of an n<\/span>th order differential equation is one involving n<\/span> (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 $y'' = 2$ . <\/p>\n
A particular solution<\/span> 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 c_{1<\/sub><\/span> = -3 and c_{2<\/sub><\/span> = 2. <\/p>\n}}
A singular solution<\/span> 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<\/span>. This second solution is a singular solution. <\/p>\n
Differential equation of a family of curves<\/h2>\n
A general solution of an n<\/span>th order differential equation has n<\/span> arbitrary constants (or parameters) and represents geometrically an n parameter family of curves<\/span>. Conversely a relation with n<\/span> arbitrary constants (sometimes called a primitive<\/span>) has associated with it a differential equation of order n<\/span> (of which it is a general solution) called the differential equation of the family<\/span>. This differential equation is obtained by differentiating the primitive n<\/span> times and then eliminating the n<\/span> arbitrary constants among the n<\/span> + 1 resulting equations. <\/p>\n<\/blockquote>\n
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Order of a differential equation<\/h2>\nAn equation having a derivative of n<\/span>th order but no higher is called an n<\/span>th order differential equation<\/span>. <\/p>\n

Order of a differential equation<\/h2>\n
An equation having a derivative of n<\/span>th order but no higher is called an n<\/span>th order differential equation<\/span>. <\/p>\n