EXAMPLES: THE LAME-EMDEN ORDINARY DIFFERENTIAL EQUATION (ODE) FOR
POLYTROPIC STELLAR INTERIORS
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The general Lane-Emden ODE (y is mass density divided by mass density 
at the center of the star; x is distance from the center of the star
multiplied by a scaling factor):

       y''(x) + 2y'(x)/x + y^n = 0

Symbol "^" means y is raised to the power n.

Boundary conditions:

        y(0) = 1
       y'(0) = 0

The parameter n is defined from the relation

        GAMMA = 1+1/n,  where Pressure = Constant * Density^GAMMA





(1) EXAMPLE #1 - n=1.5 polytrope (i.e., GAMMA = 5/3):

In the runge.f ODE solver program, try the following:

Domain of integration, etc.:
NUM = 10000
A = 0.00001  <== note: x yields a discontinuity at zero for this ODE
B = 15
SKIP = 10

Initial boundary conditions:
U(1) = 0   <== i.e., value of y'(x) at beginning
U(2) = 1   <== value of y(x) at beginning

Function declarations:
F1 = -2*U1/X - U2*SQRT(ABS(U2))
F2 = U1
F3 = 0
F4 = 0
   .
   .
   .
F10 = 0

Meaningful values for the solution of y(x) are for x=0 extending
to the first zero value for y(x) (maximum x value is about 3.65).





(2) EXAMPLE #2 - n=3 polytrope (i.e., GAMMA = 4/3):

Domain of integration, etc.:
NUM = 10000
A = 0.00001
B = 15
SKIP = 10

Initial boundary conditions:
U(1) = 0
U(2) = 1

Function declarations:
F1 = -2*U1/X - U2*U2*U2
F2 = U1
F3 = 0
F4 = 0
   .
   .
   .
F10 = 0

Meaningful values for the solution of y(x) are for x=0 extending
to the first zero value for y(x) (maximum x value is about 6.90).