From: "Steve Olson (RJG)" Subject: Re: How to solve this ODE To: rusin@math.niu.edu (Dave Rusin) Date: Fri, 24 Mar 1995 11:15:54 -0600 (CST) Dave Rusin writes: > > In article you write: > >Could some kind person help get me started in the right direction to > >find an analytical solution to this ODE? > > > > y'' - c x^(-1/4) y = 0 > > > >where ' denotes differentiation wrt x. > > I'd guess there is no analytic solution, but I'd be pleased to hear of > anything you learn. Best suggestion I could make is to transform it into > a non-linear first-order ODE by letting u = y'/y, so that u'=y''/y-(y'/y)^2 > and you have u'+ u^2 = c x^(-1/4). I tried a large number of substitutions > by did not succeed in making this look any nicer. > > Of course as a linear ODE is quite amenable to numeric solutions. > > dave Thanks for trying Dave, I could not find a good way to transform it either, but I did get some help from two others if you're interested. I've appended their replies below. ------- From: "Zdislav V. Kovarik" To: sjo@me.umn.edu Subject: Re: How to solve this ODE In article you write: >Hi all, > >Could some kind person help get me started in the right direction to >find an analytical solution to this ODE? > > y'' - c x^(-1/4) y = 0 > >where ' denotes differentiation wrt x. > >Thanks, Steve Problems of this kind have solutions of the form y = x^a * J_b (k*x^p) where J_b is a Bessel function of index b, and the parameters a, b, k, p are to be matched to make J satisfy Bessel differential equation t^2 * J''(t) + t * J'(t) + (t^2 - b^2) J(t) = 0 (the change of variable being t = k * x^p). Two independent solutions are obtained, and the general solution follows. If b comes out as integer, the other independent solution involves a Hankel function Y of the appropriately changed variable. A more detailed source of information might be a handbook on Bessel functions, or in general, a good handbook on special functions. Hope it helps, ZVK (Slavek). ------- From chen@fractal.eng.yale.edu Fri Mar 17 17:56:53 1995 To: sjo@me.umn.edu Subject: Re: How to solve this ODE This is obtained using Mathematica In[4]:= DSolve[{y''[x]-c/x^(1/4) y[x]==0},y[x],x] c y[x] Out[4]= DSolve[{-(------) + y''[x] == 0}, y[x], x] 1/4 x In[5]:= Needs["Calculus`DSolve`"] In[6]:= %4 Out[6]= {{y[x] -> -16 c 2/7 4 -16 c 7/8 > (-----) Sqrt[x] BesselJ[-(-), 2 Sqrt[-----] x ] C[1] + 49 7 49 -16 c 2/7 4 -16 c 7/8 > (-----) Sqrt[x] BesselJ[-, 2 Sqrt[-----] x ] C[2]}} 49 7 49 Good Luck -- Richard Q. Chen chen@fractal.eng.yale.edu ------- Thanks again, Steve -- Steven J. Olson Department of Mechanical Engineering University of Minnesota Email: sjo@me.umn.edu 111 Church St. SE Voice: 612-625-6634 Minneapolis, MN 55455