From: israel@math.ubc.ca (Robert Israel) Subject: Re: dy/dx=A*x^b+1/y Date: 18 Oct 2001 22:58:34 GMT Newsgroups: sci.math Summary: Abel equation (first-order ordinary differential equation) In article , Maxim wrote: >Does somebody know how to solve the differential equation >dy/dx=A*x^b+1/y? According to Maple, this is an Abel equation of the second type and class A. Maple can't find a closed-form solution, even in the special case b=2. In the case b=1 it can find a solution in terms of Airy functions: (1/3) (2/3) (1/3) (2/3) (_C1 2 A x AiryBi(%1) + I _C1 2 A x AiryBi(%1) sqrt(3) (1/3) (2/3) + 4 _C1 AiryBi(1, %1) + 2 A x AiryAi(%1) (1/3) (2/3) / + I 2 A x AiryAi(%1) sqrt(3) + 4 AiryAi(1, %1)) / ( / (1/3) (2/3) (1/3) (2/3) 2 A x AiryBi(%1) + I 2 A x AiryBi(%1) sqrt(3) + 4 AiryBi(1, %1)) = 0 (1/3) 2 (2/3) 2 %1 := 1/16 A (A x - 2 y(x)) 2 (1 + I sqrt(3)) Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2