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Re: Methods of solving nonlinear equation
rdm at hpbs2500.boi.hp.com (Bob Morrison) wrote: > >I'm a non-expert with Mathematica and mathematics, so this is probably >kind of a stupid question: when I tried to solve a common nonlinear >equation D^2 y[t] + k/(y[t]^2) == 0, I seem to be unable to get solutions. >On paper, there is a degenerate solution at (a +/-b t)^(2/3), and >also on paper, you can derive a polynomial solution (a0 + a1 t + > a2 t^2 + ....) which shows two degrees of freedom (a0 and a1 are >independently unconstrained). However, the polynomial solution's >region of convergence is too small to be very useful to me, and the >boundary conditions of my problem disallow the degenerate solution. > I tried using Mathematica to get solutions using DSolve and NDSolve, >but am too inexperienced to get anything useful. It turns out if you >assume that y is of form (f(t))^(2/3) you get another nonlinear equation >for f(t) which appears to be just as unsolvable. I tried representing >y in Exp[f(t)] form, and failed here. I tried separation >of variables and didn't get anywhere (to be honest, I can't remember >why, it's been a little while since I tried it). Can anybody >provide a reference or maybe a little hint how to do this? Ideally, >I would like a closed form solution, of course, but a limiting function >would also be useful. I created a difference equation which correctly >shows a coarse version of the solution, but in the region of interest, >the dt portion is too large to correctly solve the problem. I think >the solutions will involve an elliptic integral, but I'm just guessing >here. I freely admit this is not my area of expertise and may have >overlooked the obvious solution, but I don't see it... > >Thanks, Bob Morrison, rdm at hpbs670.boi.hp.com > Hi Bob. I think the standard trick for doing something like y'' + k / y^2 = 0 is to multiply both sides by y' and then integrate w.r.t. x. This gives you something like y' = [ c + 2 k / y ]^(1/2). (c is just an integration constant.) Rearranging, you get x = x0 + Integrate[ 1 / Sqrt[ c + 2 k / y ], y], which you can either do by hand or have mma do for you. This will prob. give you some kind of trancendental relationship betw. x and y which you can look at in whatever limit you desire. The solution will be parametrized by the two integration constants, x0 and c. Hope this helps. -Brian -- Brian J. Albright | Department of Physics and Astronomy, UCLA | To err is human... albright at physics.ucla.edu | to err really big http://bohm.physics.ucla.edu/~albright | is government.