Difficulty with NDSolve (and DSolve)

*To*: mathgroup at smc.vnet.net*Subject*: [mg106125] Difficulty with NDSolve (and DSolve)*From*: KK <kknatarajan at yahoo.com>*Date*: Sat, 2 Jan 2010 05:04:58 -0500 (EST)

Hi, I am trying to numerically solve a differential equation. However, I am encountering difficulty with Mathematica in generating valid numerical solutions even for a special case of that equation. The differential equation for the special case is: F'[x] == - (2-F[x])^2/(1-2 x + x F[x]) and F[1]==1. These equations are defined for x in (0,1). Moreover, for my context, I am only interested in solutions with F[x] in the range <1. Even before I used Mathematica, I had computed the solution to the differential equation as the solution to the following : x (2-F[x]) - Log[2-F[x]]==1. For any given x in (0,1), there are two values of F [x] that satifsy the equation. One of them is always less than 1, and the other is F[x]= 2 + (ProductLog[E^(I (I + \[Pi])) x])/x which is always greater than 1. For example, when x=0.85, the solutions are F [x]=0.2979 and F[x]=1.32407. As mentioned earlier, I am only interested in the first solution. Both DSolve and NDSolve seem to provide only the second solution and not the first one. When using DSolve, it gives out a warning that there may be multiple solutions but that's about it. Is there an option or something that I can set with NDSolve (or even DSolve) to generate the solutions of interest to me? Below is the relevant set of Mathematica code and the corresponding outputs. I would greatly appreciate your help in this regard. Thanks, KK ------------------ In[1]:= DSolve[{ -((-2 + F[x])^2/(1 - 2 x + x F[x])) == F'[x], F[1] == 1}, F[x], x] During evaluation of In[1]:= InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses. >> During evaluation of In[1]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >> During evaluation of In[1]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >> Out[1]= {{F[x] -> (2 x + ProductLog[E^(I (I + \[Pi])) x])/x}} ------------------ (*Sample solution*) In[2]:= Solve[(1 == (x (2 - F[x]) - Log[2 - F[x]] /. F[x] -> F) /. x -> 0.9), F] During evaluation of In[2]:= InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses. >> During evaluation of In[2]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >> Out[2]= {{F -> 0.297987}, {F -> 1.32407}} -------------------- (*Please note I am using F[0.9999]=1 as the initial condition in the NDSolve below. \ Otherwise, I end up with a warning that "Infinite expression 1/0. \ encountered. I had also used the same initial condition with DSolve \ and the results were similar.*) In[3]:= NumSol = NDSolve[{ -((-2 + F[x])^2/(1 - 2 x + x F[x])) == F'[x], F[.9999] == 1}, F, {x, 0, 1}] During evaluation of In[3]:= NDSolve::ndsz: At x == 0.9999000049911249`, step size is effectively zero; singularity or stiff system suspected. >> Out[156]= {{F -> \!\(\* TagBox[ RowBox[{"InterpolatingFunction", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"0.`", ",", "0.9999000049911249`"}], "}"}], "}"}], ",", "\<\"<>\"\>"}], "]"}], False, Editable->False]\)}} --- (* I have omitted the plot here but it generates only the second result*) In[157]:= Plot[{Evaluate[F[x] /. NumSol]}, {x, .33, 1}] -----------------------

**Follow-Ups**:**Re: Difficulty with NDSolve (and DSolve)***From:*danl@wolfram.com

**Re: Difficulty with NDSolve (and DSolve)***From:*Mark McClure <mcmcclur@unca.edu>