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Integration of some very elementary functions

Jan Mařík — 1993

Mathematica Bohemica

Let m be a natural number. Let f , g and Q be real polynomials such that { d e g f , d e g g } ⊂ { 1 , 2 } , d e g Q < m d e g f , g is not a square and f has imaginary roots, if it is not linear. Effective methods for the integration of Q / ( f m g are exhibited.

Integrating factor

Jan Mařík — 1994

Mathematica Bohemica

The problem of integrating factor for ordinary differential equations is investigated. Conditions are given which guarantee that each solution of ∂ 1 F ( x , y ) + y ' ∂ 2 F ( x , y ) = 0 is also a solution of M ( x , y ) + y ' N ( x , y ) = 0 where ∂ 1 F = μ M and ∂ 2 F = μ N .

A note on integration of rational functions

Jan Mařík — 1991

Mathematica Bohemica

Let P and Q be polynomials in one variable with complex coefficients and let n be a natural number. Suppose that Q is not constant and has only simple roots. Then there is a rational function ϕ with ϕ ' = P / Q n + 1 if and only if the Wronskian of the functions Q ' , ( Q 2 ) ' , ... , ( Q n ) ' , P is divisible by Q .

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