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General definition

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The statement of the theorem

Given FF , a function of xx , yy , and derivatives of yy . Then an equation of the form F(x,y,y′,…,y(n−1))=y(n)F\left(x,y,y',\ldots ,y^{(n-1)}\right)=y^{(n)} is called an explicit ordinary differential equation of order nn . More generally, an implicit ordinary differential equation of order nn takes the form: F(x,y,y′,y′′, …, y(n))=0F\left(x,y,y',y'',\ \ldots ,\ y^{(n)}\right)=0 There are further classifications: Autonomous A differential equation is autonomous if it does not depend on the variable x.Linear A differential equation is linear if FF can be written as a linear combination of the derivatives of yy ; that is, it can be rewritten as y(n)=∑i=0n−1ai(x)y(i)+r(x)y^{(n)}=\sum _{i=0}^{n-1}a_{i}(x)y^{(i)}+r(x) where ai(x)a_{i}(x) and r(x)r(x) are continuous functions of xx . The function r(x)r(x) is called the source term, leading to further classification. Homogeneous A linear differential equation is homogeneous if r(x)=0r(x)=0 . In this case, there is always the "trivial solution" y=0y=0 .Nonhomogeneous (or inhomogeneous) A linear differential equation is nonhomogeneous if r(x)≠0r(x)\neq 0 . Non-linear A differential equation that is not linear. - ^ ^{a}^{b}Harper (1976, p. 127) - ^Kreyszig (1972, p. 2) - ^Simmons (1972, p. 3) - ^ ^{a}^{b}Kreyszig (1972, p. 24) - ^Simmons (1972, p. 47) - ^Harper (1976, p. 128)