Trying to Solve system of 2 Equations "Unable to find explicit solution" Code In Description

Code pasted below, thanks in advance:

clear all;
close all;
clc;

% Solve System of Equations


syms a1 a2 a3 r_1 r_2 alpha E

a3 = (alpha*E)/(8*pi);

[sol_a1, sol_a2] = solve(2*a3*log(r_1) + (a1/(r_1)^2) + a3 + 2*a2 == 0,...
    2*a3*log(r_2) + (a1/(r_2)^2) + a3 + 2*a2 == 0);

Warning: Unable to find explicit solution. For options, see help.

Answer 1

[Walter Roberson]

syms a1 a2 a3 r_1 r_2 alpha E
a3 = (alpha*E)/(8*pi);
eqn = [2*a3*log(r_1) + (a1/(r_1)^2) + a3 + 2*a2 == 0,...
    2*a3*log(r_2) + (a1/(r_2)^2) + a3 + 2*a2 == 0];

partial_1 = solve(eqn(1), a1)

partial_1 = 

eqn2 = subs(eqn(2:end), a1, partial_1)

eqn2 = 

partial_2 = solve(eqn2(1), a2)

partial_2 = 

sol_a2 = partial_2

sol_a2 = 

sol_a1 = subs(partial_1, a2, sol_a2)

sol_a1 = 

%cross-check
subs(eqn, {a1, a2}, {sol_a1, sol_a2})

ans = 

isAlways(ans)

ans = 1×2 logical array
   1   1

%or... more directly...
syms a1 a2 a3 r_1 r_2 alpha E
a3 = (alpha*E)/(8*pi);
[sol_a1, sol_a2] = solve(2*a3*log(r_1) + (a1/(r_1)^2) + a3 + 2*a2 == 0,...
    2*a3*log(r_2) + (a1/(r_2)^2) + a3 + 2*a2 == 0, [a1, a2])

sol_a1 = 

sol_a2 = 

simplify(sol_a1)

ans = 

simplify(sol_a2)

ans = 

[Walter Roberson]

When you ask to solve() a system of equations, and you do not pass a list of variables to solve for, then solve() will use symvar() to find the list of "free variables" in the expression, and then will pick the first however-many out of the list according to the number of equations. But the "first" variables returned by symvar() might not happen to be anything to do with what you were hoping to solve for.

symvar() has a specific order that it returns variables in, but it is difficult to find documentation of that order these days. In particular solve() calls it in an undocumented way, and the logic of the sorting is deep inside the symbolic engine. The order is xywzvutsrqponmlkjihgfedcbaXYWZVUTSRQPONMLKJIHGFEDCBA . Notice that lower-case r is earlier in the order than lower-case a, so it is your r_1 and r_2 variables that are picked out to solve for.

[Michael Meyne]

Thank you, very helpful!

[Michael Meyne]

I have a quick question, how do you get the solution to be printed with this formatting?

[Walter Roberson]

@Michael Meyne

When you have a sym() or symfun() or symmatrix() expression, then that sort of formatting happens automatically if you are using LiveScript or MATLAB Online or MATLAB Answers. It does not happen if you are using the traditional command window (and there is no way to turn it on for the command window)

Answer 2

[Sulaymon Eshkabilov]

Note that you have six unknowns (a1, a2, r_1, r_2, alpha, E) an two equations. It is possible to get two solution type expressions using IgnoreAnalyticalConstraints option, e.g.:

syms a1 a2 r_1 r_2 alpha E

a3 = (alpha*E)/(8*pi);

SOLUTION = solve(2*a3*log(r_1) + (a1/(r_1)^2) + a3 + 2*a2 == 0,...
    2*a3*log(r_2) + (a1/(r_2)^2) + a3 + 2*a2 == 0, "IgnoreAnalyticConstraints",true)

SOLUTION = struct with fields:
    r_1: exp(-((E*alpha)/2 + 8*pi*a2)/(E*alpha))*exp(wrightOmega(log(-(8*pi*a1)/(E*alpha)) + (E*alpha + 16*pi*a2)/(E*alpha))/2)
    r_2: exp(-((E*alpha)/2 + 8*pi*a2)/(E*alpha))*exp(wrightOmega(log(-(8*pi*a1)/(E*alpha)) + (E*alpha + 16*pi*a2)/(E*alpha))/2)

r_1 = SOLUTION.r_1

r_1 = 

r_2 = SOLUTION.r_2

r_2 = 

a3 = (alpha*E)/(8*pi)

a3 =