kill(all)$

Elektrochemistry :appendix1 (diff(ψ,x,2) = b*sinh(ψ),version 1.0)
peter.vlasschaert@gmail.com,10/03/2019

load packages.

load(contrib_ode)$
load(lrats)$

why : load(contrib_ode) when 'ode2'  , gives no solution for the problem.

p1:'diff(ψ,x,2) =sinh(ψ);

depends(ψ,x);

p1a:ode2(p1,ψ,x);

rem : [- f(ψ) -x -%k2 = 0,  f(ψ)-x-%k2 =0]

both parts of solution : because sqrt 
===============================

p1b:p1a[1];
p1c:p1a[2];

cosh(ψ)  , definition.
=================

p2:cosh(ψ)  = exponentialize(cosh(ψ));

what about integral.  
===========================
integrate(f(ψ),ψ(%k1))  = x + %k2

How to find : which to use  'p1b' or 'p1c' ?
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general  :
========

p3:integrate(1/sqrt(cosh(ψ)+%k1),ψ);

p3a: rhs(p1c);

calculus : courses integration of  hyperbolic function .

rem: theory understand the result
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p4: 'integrate(f(ψ),ψ)  = x + const;

p4a:diff(p4);

1e) del(const)  = 0, always zero .
===========================

p4c :f(ψ)*del(ψ)  = del(x);

2e) translation : mathematics .
=========================

p4d:diff(ψ,x) = f(ψ);

%k1,%k2  ?  Found from BC ( = (B)oundary (C)ondition on the domain 
specefied by the problem )
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example :   ψ            = a         when  x = infinity =∞
                     dψ/dx    = c         when  x = infinity =∞
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rem : a,c = real number

assume

assume(b>0);

rem : Why otherwise very difficult solution,for a  general :  'b value'

p5:'diff(ψ,x,2) =b*sinh(ψ);
p5a:ode2(p5,ψ,x);

relation :easy to see relation between 'p1a' and 'p5a'.

analysis : problem
-----------------------------

p6:p1;

rem :  b =1  (because assume , b > 0).

lhs : 'p6'
=======

p6a:diff(ψ,x) = p;

p6a1: rhs(p6a) = lhs(p6a);

p6b:'diff(ψ,x,2) = 'diff(rhs(p6a),x);

chance rule :

p6c:lhs(p6b)  = diff(ψ,x)*'diff(p,ψ);

p6d:subst([lhs(p6a) = rhs(p6a)],p6c);

result  'p6'
=========

p7:rhs(p6d)  = rhs(p6);

mathematics =>  dψ to the right.
===========================

p7a: 'integrate(p,p) = 'integrate(rhs(p6),ψ) + %k3;

p7b:ev(p7a,nouns);

rem : integration : hyperbolic sin  =>  hyperbolic cos.

How to solve : 'p7b'
------------------------------

p8:subst([p6a1],p7b);

p8a:solve(p8,'diff(ψ,x,1));

rem : two  'ODEs'
--------------------------

p8a1:p8a[1];
p8a2:p8a[2];

solve : 'p8a1'
--------------------

p9:ode2(p8a1,ψ,x);

p9a:subst([%k3=%k1,%c=%k2],p9);

rem : same result  'p1b'

solve : 'p8a2'
--------------------

p10:ode2(p8a2,ψ,x);

p10a:subst([%k3=%k1,%c=%k2],p10);

rem : same result  'p1c'

solution problem, 'p9a','p10a' : select  'on the physics of the problem'.

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