kill(all)$

Elektrochemistry :appendix1 :Δψ = ψ (poisson (radial),version 1.0)
peter.vlasschaert@gmail.com,08/03/2019

see:Elektrochemistry :appendix :Δψ (spherical coordinates Δψ(r),version 1.0)
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simplified : ψ(r,θ,φ)   = >  ψ (r) = y(r)   and  α = 2   'see below '
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load packages
============

load(contrib_ode);
load(lrats);

rem : 'ode2 ' give no solution for  'p1' , see below.

assume(r≥0);

rem : cartesian coordinates :  Δ(ψ)
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ψ = ψ (x,y,z)  :  3D
ψ  = ψ (x,y)    :  2D
ψ =  ψ (x)        : D
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3D :  examples (PDE)
==================

p11a :Δ(ψ)  = 'diff(ψ,x,2) + 'diff(ψ,y,2)+ 'diff(ψ,z,2);

DEF : laplace equation 
===================

p11b: rhs(p11a) = 0;

DEF : poisson equation
====================

p11c:rhs(p11a)  =  f(x,y,z);

α = 1 : cylindrical coordinates ( only radial distribution)
α = 2 : spherical   coordinates ( only radial distribution)
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pde : poisson equation Δ y(r)  = y(r) .
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rem : cartesian coordinates : 

change values : α  'see above'
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parametric : α
============

α:2  ;

p1:1/r^α*'diff(r^α*'diff(y,r),r) = 'y;

depends(y,r);

p1a:ev(p1,nouns),ratexpand;

use :  ode2   (solve ODE)
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p1b1: ode2(p1a,y,r);

→' no solution'

use :  contrib_ode   (solve ODE)
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p1b2:contrib_ode(p1a,y,r);

→ ' solution '

procedure : check your  answer ( from  'p1b2')
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put  'p1b2' into 'p1a'
=================

p1c:subst(p1b2,p1a);

ev( ......,diff)
ev(........,nouns)
=============

p1d:ev(p1c,nouns)$

simplify
=======

p1e:ratexpand(p1d);

check :equation.  'p1a'  with 'p1b2'
=============================

p1f:is(p1e);

%k1, %k2  can be found  from BC = (B)oundary (C)onditions. (general : PDE)
see  : problem in question.
================================================================
PDE : (p)artial (d)ifferential (e)quations.      '   examples  :  p11b,p11c '
ODE : (o)rdinary (d)ifferential (e)quations.  '   example    :  p1a '
================================================================