{"id":4529,"date":"2013-12-28T06:05:05","date_gmt":"2013-12-27T21:05:05","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4529"},"modified":"2014-08-01T19:03:04","modified_gmt":"2014-08-01T10:03:04","slug":"ordinary-differential-equations","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4529","title":{"rendered":"Ordinary differential equations"},"content":{"rendered":"
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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, c1<\/sub>, c2<\/sub><\/span>, etc., may assume values independently of the variables involved. For example in y = x^2 + c_1x + c_2, c1<\/sub><\/span> and c2<\/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 c1<\/sub><\/span> = -3 and c2<\/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