Reduction of order
Reduction of order [ edit ] Differential equations can usually be solved more easily if the order of the equation can be reduced. Reduction to a first-order system [ edit ] Any explicit differential equation of order n , � ( � , � , � ′ , � ″ , … , � ( � − 1 ) ) = � ( � ) can be written as a system of n first-order differential equations by defining a new family of unknown functions � � = � ( � − 1 ) . for i = 1, 2,..., n . The n -dimensional system of first-order coupled differential equations is then � 1 ′ = � 2 � 2 ′ = � 3 ⋮ � � − 1 ′ = � � � � ′ = � ( � , � 1 , … , � � ) . more compactly in vector notation: � ′ = � ( � , � ) where � = ( � 1 , … , � � ) , � ( � , � 1 , … , � � ) = ( � 2 , … , � � , � ( � , � 1 , … , � � ) ) . Summary of exact solutions [ edit ] Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given here. In the table below, P (...