According to Maple, this is the solution:
> EQ:=diff(y(x),x,x)+2*y(x)*diff(y(x),x)-x=0;
/ 2 \
|d | /d \
EQ := |--- y(x)| + 2 y(x) |-- y(x)| - x = 0
| 2 | \dx /
\dx /
> dsolve(EQ);
1/2 1/2 2
_C1 2 2 x
2 _C2 WhittakerW(-------- + 1, 1/4, -------)
4 2
y(x) = - --------------------------------------------
x (_C2 %2 + %1)
1/2 1/2 2
1/2 _C1 2 2 x
(3 + _C1 2 ) WhittakerM(-------- + 1, 1/4, -------)
4 2
+ 1/2 ----------------------------------------------------- +
x (_C2 %2 + %1)
2 1/2 1/2 1/2 2 1/2
(-_C2 + _C2 x 2 - _C2 2 _C1) %2 + (-1 + 2 x - _C1 2 )
%1
1/2
---------------------------------------------------------------------
x (_C2 %2 + %1)
1/2 1/2 2
_C1 2 2 x
%1 := WhittakerM(--------, 1/4, -------)
4 2
1/2 1/2 2
_C1 2 2 x
%2 := WhittakerW(--------, 1/4, -------)
4 2
Where, as per Maple Help, the Whittaker functions...
can be defined in terms of the hypergeometric and Kummer functions as
follows:
(1/2 + nu)
WhittakerM(mu, nu, z) = exp(- 1/2 z) z
hypergeom([1/2 + nu - mu], [1 + 2 nu], z)
(1/2 + nu)
WhittakerW(mu, nu, z) = exp(- 1/2 z) z
KummerU(1/2 + nu - mu, 1 + 2 nu, z)
Maple Help gives the following references:
Abramowitz, M. and Stegun I. Handbook of Mathematical Functions. Chapter 13.
Luke, Y. The Special Functions and Their Approximations. Volume 1. Chapter
4.
No, definitely not something I'd expect ODE2() to solve.
Viktor
-----Original Message-----
From: maxima-admin@math.utexas.edu [mailto:maxima-admin at math] On
Behalf Of Billinghurst, David (CALCRTS)
Sent: Tuesday, February 22, 2005 6:03 PM
To: Juan Pablo Romero Bernal; maxima@math.utexas.edu
Subject: RE: [Maxima] Problem with differential equation!!!
> I want resolve a simple differential equation,
>
> 2
> d y + 2y dy -x = 0
> --- ---
> 2
> dx dx
>
I had a quick look through Kamke[1] and Murphy[2] and didn't see
a solution. It is possible that no closed form solution exists.
David
[1] E Kamke, Differentialgleichungen Losungsmethoden und Losungen,
Vol 1, Geest & Portig, Leipzig, 1961
[2] G M Murphy, Ordinary Differential Equations and Their Solutions,
Van Nostrand, New York, 1960
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