michaelvio

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Excellent answer but something is not ok is entirely my fault the equation is rH''(r)+H'(r)+(rk^2-r^2*b^2/R^2)H(r)=0 where k, b, and R are real constant positive number, with condition H(R)=0 and H'(1/R)=R to be solved into series of power. The SECOND CONDITION IS TO DERIVATE H'(1/R)=R that's why I'm interested in an approximate solution based on series, or any results as long as it satisfied the too condition H(R)=0 and H(1/R)=R of the real function H(r) and if is possible a plot for R=370, k=100 and b=35. Very nice I learn a lot about it! could you add an extra help, please!

Sorry, the equation is rH''(r)+H'(r)+(rk^2-r^2*b^2/R^2)H(r)=0 where k, b, and R are real constant positive number, with condition H(R)=0 and H(1/R)=R to be solved into series of power. I know from the literature that xy''+y'+xy=0, can't be solved in terms of elementary function(see G.Nagy-ODE-November 29, 2017) that's why I'm interested in an approximate solution based on series, or any results as long as it satisfied the too condition H(R)=0 and H(1/R)=R of the real function H(r) and if is possible a plot for R=370, k=100 and b=35.

Please advice!.

ok asympt is a good command!

asympt(G, r):

but I'm also interested in a general solution where R= an arbitrary constant, not a value!

 

@vv 

I'm very interested in an approximate solution but unfortunately is not correct you put R=0 and convert  it into series 

g1:=convert(series(%, R=0), polynom);  

I have my own workout 
 

restart

with(Slode):

ode := diff(diff(g(r), r), r)-r*g(r)/R = 0

diff(diff(g(r), r), r)-r*g(r)/R = 0

(1)

Order := 10;

10

(2)

dsolve({diff(g(r), r, r)-r*g(r)/R = 0, g(2*R) = 0}, g(r), series);

g(r) = series((D(g))(2*R)*(r-2*R)+((1/3)*(D(g))(2*R))*(r-2*R)^3+((1/12)*(D(g))(2*R)/R)*(r-2*R)^4+((1/30)*(D(g))(2*R))*(r-2*R)^5+((1/60)*(D(g))(2*R)/R)*(r-2*R)^6+((1/2520)*(D(g))(2*R)*(4*R^2+5)/R^2)*(r-2*R)^7+((1/840)*(D(g))(2*R)/R)*(r-2*R)^8+((1/45360)*(D(g))(2*R)*(2*R^2+13)/R^2)*(r-2*R)^9+O((r-2*R)^10),r = 2*R,10)

(3)

FPseries(ode, g(r), a(n), 'free' = A, 'terms' = 9);

FPSstruct(A[0]+A[1]*r+(1/6)*A[0]*r^3/R+(1/12)*A[1]*r^4/R+(1/180)*A[0]*r^6/R^2+(1/504)*A[1]*r^7/R^2+(1/12960)*A[0]*r^9/R^3+Sum(a(n)*r^n, n = 10 .. infinity), (R*n^2-R*n)*a(n)-a(n-3))

(4)

g := proc (r) options operator, arrow; A[0]+A[1]*r+(1/6)*A[0]*r^3/R+(1/12)*A[1]*r^4/R+(1/180)*A[0]*r^6/R^2+(1/504)*A[1]*r^7/R^2+(1/12960)*A[0]*r^9/R^3 end proc;

proc (r) options operator, arrow; A[0]+A[1]*r+(1/6)*A[0]*r^3/R+(1/12)*A[1]*r^4/R+(1/180)*A[0]*r^6/R^2+(1/504)*A[1]*r^7/R^2+(1/12960)*A[0]*r^9/R^3 end proc

(5)

solve([g(2*R) = 0, (D(g))(0) = R], {A[0], A[1]});

{A[0] = -(90/7)*R^2*(8*R^4+42*R^2+63)/(16*R^6+144*R^4+540*R^2+405), A[1] = R}

(6)

subs(A[0] = -(90/7)*R^2*(8*R^4+42*R^2+63)/(16*R^6+144*R^4+540*R^2+405), A[1] = R, g(r));

-(90/7)*R^2*(8*R^4+42*R^2+63)/(16*R^6+144*R^4+540*R^2+405)+R*r-(15/7)*R*(8*R^4+42*R^2+63)*r^3/(16*R^6+144*R^4+540*R^2+405)+(1/12)*r^4-(1/14)*(8*R^4+42*R^2+63)*r^6/(16*R^6+144*R^4+540*R^2+405)+(1/504)*r^7/R-(1/1008)*(8*R^4+42*R^2+63)*r^9/(R*(16*R^6+144*R^4+540*R^2+405))

(7)

g := proc (r) options operator, arrow; -(90/7)*R^2*(8*R^4+42*R^2+63)/(16*R^6+144*R^4+540*R^2+405)+R*r-(15/7)*R*(8*R^4+42*R^2+63)*r^3/(16*R^6+144*R^4+540*R^2+405)+(1/12)*r^4-(1/14)*(8*R^4+42*R^2+63)*r^6/(16*R^6+144*R^4+540*R^2+405)+(1/504)*r^7/R-(1/1008)*(8*R^4+42*R^2+63)*r^9/(R*(16*R^6+144*R^4+540*R^2+405)) end proc;

proc (r) options operator, arrow; -(90/7)*R^2*(8*R^4+42*R^2+63)/(16*R^6+144*R^4+540*R^2+405)+R*r-(15/7)*R*(8*R^4+42*R^2+63)*r^3/(16*R^6+144*R^4+540*R^2+405)+(1/12)*r^4-(1/14)*(8*R^4+42*R^2+63)*r^6/(16*R^6+144*R^4+540*R^2+405)+(1/504)*r^7/R-(1/1008)*(8*R^4+42*R^2+63)*r^9/(R*(16*R^6+144*R^4+540*R^2+405)) end proc

(8)

R := 3.7*10^2;

370.0

(9)

sort(g(r), r, ascending);

-6.428395340+370.0*r-0.2895673577e-2*r^3+(1/12)*r^4-0.2608714934e-6*r^6+0.5362505363e-5*r^7-0.9792473478e-11*r^9

(10)

plot(g(r), r = 5*10^21 .. .9*10^22);

 

DG := diff(g(r), r);

370.0-0.8687020731e-2*r^2+(1/3)*r^3-0.1565228960e-5*r^5+0.3753753754e-4*r^6-0.8813226130e-10*r^8

(11)

plot(DG(r), r = 5*10^21 .. .9*10^22);

 

``

but I didn't resolve in a recurrent mod with the expression of the general term.
Please advice!

Download SloNew2.mw

 

 Does Maple 2019 or 2017 come with significant improvement in solving this kind of problem? I did my self the first calculation with Airy it's very easy but I want the solution with series powers with package slode...

Thanks a lot! very useful...could I have on Email?

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