Non-homogeneous and linear-differential-equation solutions (update:13-07-07)

homogen and non-homogen solution

DESCRIPTION;

This program is a running module for homsolution.m Matlab-functions. Also, differential non-homogeneous or homogeneous equations are solution possible the Matlab&Mapple Dsolve.m&desolve main-functions. But;

EXAMPLE;
[1]--+---Sometime mapple function is produce more short solution
|--- My function's solution:

[ R^4-4*R^3 ]*(y) = [5 ]
y = [ +exp^(4x).(C4)+exp^(0x).(C1+C2*x^1+C3*x^2) ]g + [-5/24*x^3-5/32*x^2-5/64*x-5/256 ]s
Generally solution Special solution
#### true ##### #### true #####
|
|--- Mapple's desolve function solution:

Dsolve('D4y-4*D3y-5=0','x')
ans =1/64*exp(4*x)*C1-5/24*x^3+1/2*C2*x^2+C3*x+C4
y= 1/64*exp(4*x)*C1 + 1/2*C2*x^2 + C3*x + C4 - 5/24*x^3
Generally solution Special solution (more short)
#### true ##### #### true #####

[2]---+---Matlab's Dsolve.m function is depend be selected input-veriables string character
|
|--- My function's solution:

>> homsolution([(R^4-16)^5*(R^2+1)*(R/(R^2+R+1))^2, x^20+x^10+sin(x)],0)
where R=[d/dx] and [f(R)].y = Q(x,y) differential equation solution

____Equation [1]

[ (R^4-16)^5*(R^2+1)*R^2/(R^2+R+1)^2 ]*(y) = [x^20+x^10+sin(x) ]
y= [ +exp^(-2x).(C20+C21*x^1+C22*x^2+C23*x^3+C24*x^4)+exp^(2ix)+ ...]g+ [ ...1/1518750*x*cos(x)^6+ ...]s

|
|--- Mapple's desolve function solution:

Dsolve('((Dy)^4-16)^5*((Dy)^2+1)*(Dy)^2/((Dy)^2+Dy+1)^2-(x^20+x^10+sin(x))=0','x')
(I don't advise , don't try this module, non-solution )

[3]---+---Sub-function running speed (for running 29-examples)
if you hide homsolution.m-lines(68,69,155,156) as fprintf,disp etc.. command then

My function is 1.602342 second (tic-toc & profiler control)
Matlab function is 1.094779 second

ALGORITHM;

--+--if Q(x,y)<> 0 than special solution
root value root-order degree
r1=R1 n1
r2=R2 n2
..... ....
rn=Rn nn
|---+---if root value = real
|
|----+---[max real root order degree]
else
|
Solution=1/[R-small root(1)]...
1/[R-small root(2)]...
1/[R-small root(n)]*[Q(x,y] (**)
where all step is first-order degree linear diff. equ. sol.
|---+---if root value = complex
|
|----+---[max complex root order degree]
else
|
Solution=1/[R-small root(1)]...
1/[R-small root(2)]...
1/[R-small root(n)]*[Q(x,y] (**)
where all step is first-order degree linear diff. equ. sol.

SYNTAX:

syntax.input : solution=regsolution.ouput (differential main function solution)
syntax.output: regsolt =conforming roots values for special solutions

EXAMPLE:
[ (R-2)^2*(R^2+1)^2*(R-1)^2 ]*(y) = [x^8 ]

Solution=
1.0000 2.0000
2.0000 2.0000
0 + 1.0000i 2.0000
0 - 1.0000i 2.0000

regsolt =

1.0000 firstly real roots
1.0000
2.0000
2.0000 --->look ALGORITHM<---
0 - 1.0000i
0 - 1.0000i
0 + 1.0000i secondly imaginer roots
0 + 1.0000i

Solution=[1/(R-1)][1/(R-1)][1/(R-2)][1/(R-2)][1/(R+sqrt(-1))] [1/(R+sqrt(-1))][1/(R-sqrt(-1))][1/(R-sqrt(-1))][ Q(x,y) ]