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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
>
>


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