Re: Re: Solving differential equations in the

*To*: mathgroup at smc.vnet.net*Subject*: [mg103805] Re: [mg103772] Re: [mg103768] Solving differential equations in the*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Wed, 7 Oct 2009 07:01:30 -0400 (EDT)*References*: <200910051139.HAA28779@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

No need to settle for NDSolve, when you can get the symbolic solution as follows: Clear[x, y, y1, s] Off[Solve::"ifun"] Solve the transformed ODE (thanks to Dan Dubin): x[s_] = 2 + Exp[-I s]; y1[s_] = y[s] /. First@DSolve[{y'[s]/x'[s] == Exp[y[s]], y[Pi] == 1}, y, s] -Log[-((-1 + E)/E) - E^(-I s)] (y1 is y as a function of s, not x.) Now invert the x function: s[z_] = s /. First@Solve[x[s] == z, s] I Log[-2 + z] Define y as a function of x: y[x_] = y1[s[x]] // Simplify 1 - Log[1 + E - E x] Check the initial conditions: y'[x] - Exp[y[x]] 0 y[1] 1 Bobby On Mon, 05 Oct 2009 12:16:34 -0500, Dan Dubin <ddubin at ucsd.edu> wrote: >> Hi ! >> How can I solve an ordinary differential equation of order n in the >> complex plane following a prescribed contour ? >> I can of course write my own Runge-Kutta package but is there a quickest >> way to do that (maybe NDSolve but how to define the contour ??) ? >> >> Example : NDSolve[{y'[x] == Exp[y[x]], y[1] == 1}, y, {x, 1, 3}] >> fails because of a singularity in x=1+1/e. >> However integrating the ODE following a path which avoids the >> singularity should be possible eventually leading to a multivalued >> function. >> >> Thanks for a hint. > > > Let x[s] be a complex path parametrized by a real variable s, for > instance > > x[s_] = 2 + Exp[-I s]; > > Then Re[x] runs from 1 to 3 as s runs from Pi to 0. This is the range > of x you wanted. You can choose other paths if this one is not > convenient. > > Now transform the ODE in x to an ODE in s using the chain rule, and > solve this ODE in s: > > NDSolve[{y'[s]/x'[s]==Exp[y[s]],y[Pi]==1},y,{s,0,Pi}] > > yields the function y[s] along the chosen contour. > -- DrMajorBob at yahoo.com

**References**:**Solving differential equations in the complex plane***From:*Andre Hautot <ahautot@ulg.ac.be>