Emden's Equation

# MATHEMATICAL METHODS OF Science and Engineering # FORTRAN codes / Matlab codes #http://www1.uprh.edu/rbaretti #http://www1.uprh.edu/rbaretti/Methodsoftheoreticalphysics.htm #http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicspart2.htm #http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart3.htm #http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart4.htm #http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart5.htm 

r=var('r') f(r)=r^4*exp(-r) integral(f(r),r,0,oo) 

24

24

z=var('z') f(z)=sin(z)^3 integral(f(z),z,0,pi) 

4/3

4/3

x=var('x');assume (a>0); integral(exp(-a*x^2),x,0,oo) 

1/2*sqrt(pi)/sqrt(a)

1/2*sqrt(pi)/sqrt(a)

#ap+1 = 2{ λ(p+l +1) -1} ap / [ (p+l+1)(p+l+2) - l(l+1) ] lamda=1/3 p=1 l=0 aminus=-2/3 a=2*(lamda*(p+l+1)-1)*aminus/((p+l+1)*(p+l+2)-l*(l+1) ) a 

2/27

2/27

# $ 2r2 -18r +27 }2 exp(-2r/3) 4π r2 dr r=var('r') f=(2*r^2-18*r+27)^2*exp(-2*r/3) integral(f*4*pi*r^2,r,0,oo) 

19683*pi

19683*pi

#exp(-2r) 4π r2 r=var('r') integral(exp(-2*r)*4*pi*r^2,r,0,oo) 

pi

pi

r=var('r') psi=(1/pi^(1/2))*exp(-r) integral(r*psi^2*4*pi*r^2,r,0,oo) 

3/2

3/2