How to solve this analytical equation please?

Hi,

I was trying to solve this equation to find the value of G, unfortunately could not find a way to solve it, but do not know the reason.

Any help please?

Script:

% Variables
% -----------------------------------------------------------------------
% Alpha Dash
x=0.1;

% GO
GO=1000;

% Pt
Pt=0.22e-3;

% Psat
Psat=10e-3;

% Equations
% -----------------------------------------------------------------------
a=1-x;
b=Pt/Psat;
c=1+b;
d=a*c;

% Option 1:
solve x*log(GO/G)=log(d/(a*(1+(G*b))))

% Option 2:
solve x*log(GO/G)=log((0.9*(1+b))/(0.9*(1+(G*b))))

% Option :
w= vpasolve(x*log(GO/G)=log((0.9*(1+b))/(0.9*(1+(G*b)))));

Answer 1

[Walter Roberson]

solve(x*log(GO/G)=log(d/(a*(1+(G*b)))),G)

Note: the solution involves RootOf() a 10th order equation, two of the roots of which are valid solutions.

[-15.53821708+8.146700635*i, -15.53821708-8.146700635*i]

There is no analytic solution to 10th order equations, in general.

RootOf(25937424601000 * z^10 + 11789738455000000 * z^9 + 2411537411250000000 * z^8 + 292307565000000000000 * z^7 + 23251738125000000000000 * z^6 + 1268276625000000000000000 * z^5 + 48040781250000000000000000 * z^4 + 1247812500000000000000000000 * z^3 + 21269531250000000000000000000 * z^2 + 213629777073645655956912870399 * z + 976562500000000000000000000000, z)

the 3rd and 8th roots are the valid ones.

Answer 2

[Osa]

Hi Mr Walter Roberson, Thank you for your helpful answer. Actually, I am still confused about few things and need more explanations please. I didn't get your point about "RootOf()" you mentioned and how can it be generated/used? Why did you take the 3rd and 8th roots values as valid answers? could you please clarify more? and if possible to provide the script?

[Walter Roberson]

http://www.mathworks.com/help/symbolic/mupad_ref/rootof.html

In short, RootOf(f(z),z) for some expression f(z), represents the set of numbers, z, such that f(z) = 0. The "roots" of the equation. solve() will generate RootOf structures when asked for analytic solutions to sufficiently high polynomials.

If I recall correctly, by default only polynomials of degree 0, 1, or 2 are solved directly in MuPAD, and you can request that polynomials of degree 3 or 4 be solved for algebraically (usually a real mess!); polynomials of degree 5 and higher do not usually have analytic solutions.

The sequence I used was Maple rather than MuPAD. The MuPAD notebook equivalent (not usable directly from the MATLAB prompt) would be:

Q := v -> coerce(v, DOM_RATIONAL);
x := Q(.1);
GO := 1000;
Pt := Q(0.22e-3);
Psat := Q(0.10e-1);
a := 1-x;
b := Pt/Psat;
c := 1+b;
d := a*c;
solve(x*log(GO/G) = log(d/(a*(G*b+1))), G)

Answer 3

[Parham]

if you rearrange the equation, you'll get

 a=alpha
 phi = P/Psat

 G = exp( (1/a)*(G0^a * (1-a)) / (1-a-phi+(G0^a)*a*phi) )