Re: non-linear equations not covered by built-in procedures
- To: mathgroup at smc.vnet.net
- Subject: [mg38882] Re: [mg38809] non-linear equations not covered by built-in procedures
- From: Selwyn Hollis <selwynh at earthlink.net>
- Date: Fri, 17 Jan 2003 05:38:43 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
You're making the common mistake of thinking that Mathematica is a
magic black box that can solve anything. I'm not sure what Jens-P.K.
was attempting to say about Mathematica 4.2 or solving the equation "by
hand", so let me try to say something here that humans can understand.
The appropriate technique for equations such as the one you have
(having no explicit t-dependence) is to reduce the order by substituting
r' = v and r'' = v dv/dr.
That gives you a separable equation for v in terms of y:
v dv == -1/r dr
Integration of that, and use of the initial conditions gives you v^2 =
-Log[r]. Now, since v=r' and since r'[t]<0 for small t>0 (by a
geometric argument) you have the first-order equation
r' == -Sqrt[ -2 Log[r] ]
This is not very pleasant, but it is separable. Separating and
integrating results in this "implicit" solution:
-Integrate[ 1/Sqrt[ -2 Log[y] ], {y,1,r}] == t
Mathematica will express the integral here in terms of the special
function "Erfi", but that's little more than notation. After some
massaging, you get
Sqrt[Pi/2] Erfi[Sqrt[Log[1/r]]] == t
Plot the left side over {r,0,1} and you'll see the graph of the
solution with the axes reversed.
Hope this helps.
----
Selwyn Hollis
http://www.math.armstrong.edu/faculty/hollis/mmade
On Tuesday, January 14, 2003, at 06:10 AM, Narasimham G.L. wrote:
> Hi,
>
> I am a beginner with Mathematica. Apprecite help for non-linear
> diffrl. equns.
>
> eqn = r''[t] + 1/r[t] == 0
> sol = DSolve[{eqn, r[0] == 1, r'[0] == 0}, r[t], t]
> Plot[r[t] /. sol, {t, 0, 2Pi }]
>
> carries error msg >> DSolve::dnim:
> Built-in procedures cannot solve this differential equation.
>
> The linear case when 1/r[t] is replaced by r[t] works OK,yields the
> expected Cos[t] solution.
> Is there a web accessible Mathematica reference for ODEs, non linear,
> PDEs etc.?
>
> Regards
>
>
>