Re: About Abel Type Differential Equation
- To: mathgroup at smc.vnet.net
- Subject: [mg42039] Re: [mg42018] About Abel Type Differential Equation
- From: Selwyn Hollis <selwynh at earthlink.net>
- Date: Tue, 17 Jun 2003 05:43:17 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Sergio, According to the book Intro to Nonlinear DEs and Integral Equations by Harold Davis (Dover), if we set p[x_]= (-(1/4))*(-1 + x)^(3/2)*x^2*(-1 + 27*x) a[x_]= Exp[Integral[-1/x - 3/(2(x-1)), x] b[x_]= 1/(2(x-1)) t= Integral[2(x-1)*a[x]^2,x] y[x_]= a[x]z[t] + b[x], then your equation has the canonical form D[z[t],t] == z[t]^3 + p[x] Mathematica returns a rather hideous solution to this that looks like InverseFunction[ugly x-stuff][-t/4 + C[1]] But at least it's something. Better check my set-up; I could be wrong. ----- Selwyn Hollis http://www.math.armstrong.edu/faculty/hollis On Monday, June 16, 2003, at 03:58 AM, Sergio Rojas wrote: > Hello, > > I am dealing with an Abel type differential equation that > has the form: > > eq= D[y[x],x] == 2*(x-1)*y[x]^3-3*y[x]^2-y[x]/x > > Before resorting to numerical methods, I would like to > try if I could obtain some sort of analytical solution > of that equation via Mathematica. The straight forward > application of mathematica's DSolve command, does not > work: > > DSolve[eq, y[x], x] > > However, searching Mathematica web site I came across with > this page (http://documents.wolfram.com/v4/RefGuide/DSolve_ex.html), > were apparently a more or less general form of Abel type > equation is shown (see lines corresponding to In[7] and Out[7]). > > > I am wondering if there is a pointer to a site explaining > a bit more the meaning of each one of the terms in that > equation (in line In[7]: f1, f2, f3 and > in line Out[7]: k$139, k$149, and so on). > > Sergio > >