Re: Solving differential equations?

*To*: mathgroup at smc.vnet.net*Subject*: [mg112527] Re: Solving differential equations?*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Sat, 18 Sep 2010 07:27:33 -0400 (EDT)

Zhiyong Shen wrote: > Following are the two differential equations: > > dx/dr = [a*(1-x^2+y^2)]/{r(r-1)]^(1/2)-2b*[r/(r-1)]^(1/2)*x - 0.0152*[r/(r-1)]^(1/2)*xy > > dy/dr = -2a*xy/[r(r-1)]-2b*[r/(r-1)]^(1/2)*y + 0.0076*[r/(r-1)]^(1/2) *(1+x^2-y^2) > > with initial conditions: > > x(5.67)= - {a[br + (b^2*r^2 + a^2 + 0.000058*r^2)^(1/2)]}/(a^2 + 0.000058*r^2) > > y(5.67)= -{0.0076*r[br + (b^2*r^2 + a^2 + 0.000058*r^2)^(1/2)]}/(a^2 + 0.000058*r^2) > > where r is the independent variable, x and y are dependent variables to be solved, and a and b are constant parameters. > > It would be a great help to me if someone could give me a suggestion on how to solve these equations. > > Sincerely, > > Shen I would use Mathematica. Did you try it? Because it works on this problem right out of the box. eqns = {x'[r]== (a*(1-x[r]^2+y[r]^2))/(r*(r-1))^(1/2) - 2*b*(r/(r-1))^(1/2)*x[r] - 0.0152*(r/(r-1))^(1/2)*x[r]*y[r], y'[r] == -2*a*x[r]*y[r]/(r*(r-1))-2*b*(r/(r-1))^(1/2)*y[r] + 0.0076*(r/(r-1))^(1/2)*(1+x[r]^2-y[r]^2), x[r0] == -a*(b*r0 + (b^2*r0^2 + a^2 + 0.000058*r0^2)^(1/2))/ (a^2 + 0.000058*50^2), y[r0] == -0.0076*r0*(b*r0 + (b^2*r0^2 + a^2 + 0.000058*r0^2)^(1/2))/ (a^2 + 0.000058*r0^2)}; For given parameter values (I use a=1,b=2) we can numerically solve this. {a,b} = {1,2}; In[224]:= NDSolve[eqns, {x[r],y[r]}, {r,r0,r0+10}] Out[224]= {{x[r] -> InterpolatingFunction[{{5.67, 15.67}}, <>][r], y[r] -> InterpolatingFunction[{{5.67, 15.67}}, <>][r]}} Daniel Lichtblau Wolfram Research