From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: HELP-SECOND ORDER DIFFERENTIAL EQUATION Date: 20 Jun 1999 23:55:26 -0400 Newsgroups: sci.math In article , Meetul Kinarivala wrote: :Hi All! : :I am looking for the solution of the following second order diff :equation. :Do you recognize the form ? Any hints ? Any advise on the approach ? : :y'' = 1/(Ay+c)^2 : :where A and c are known constants Yes, it is "autonomous equation with y' not appearing explicitly". You can lower the order when you multiply both sides by y' and then integrate. (Such equations come from equations of motion without friction, and the term (y')^2/2 just needs to be multiplied by the mass to become kinetic energy. Then you can solve the first order equation which is also autonomous and involves a constant of integration (total energy per unit of mass). In this case, you need not take precautions about an extra constant solution (constants do not solve the equation to start with), but watch out for "+ and -" . Even if the autonomous equation contains y' explicitly, there is an order-lowering trick: Change the variable by v = y', so that by Chain Rule y'' = dv/dx = (dv/dy) * (dy/dx) = v * dv/dy and you can re-write the equation in terms of variables v and y ("autonomous" means that x - or t, whatever, - did not show explicitly.) Then, when the resulting 1st order equation is solved (with a constant of integration kept arbitrary), you re-introduce v as dy/dx and solve to the bitter end. Good luck, ZVK(Slavek).