Solving a sine-cosine equation ( Warning: Explicit solution could not be found.)
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Hi,
I'm trying to solve a symbolic equation (the more solution I get the better is), this is my code:
clear all; close all;
syms x n a;
eq = '(n*a/2 * sin(n*a*x) * sin(a*x/2)^2 - a/2 * sin(a*x) * sin(n*a*x/2)^2) / sin(a*x/2)^4 = 0';
[S]=solve(eq,x)
and I get this message:
Warning: Explicit solution could not be found. > In solve at 83
S =
[ empty sym ]
The function eq has infinite zeros, so what's wrong in my code? Is it possible to get some solutions, let's say in the range -5<x<+5?
Any help is appreciated.
Gianluca
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Walter Roberson
2011년 7월 27일
Do you need symbolic roots? If numeric roots are acceptable then there may be mechanisms to find all the roots within a given range.
답변 (4개)
Giovanni
2011년 7월 27일
Hi Gianluca, I don't know if it's possible to limit the solutions to an interval using symbolic algebra. You might want to try solving it numerically? Also, assuming n is an integer:
equation = simple( (n*a/2 * sin(n*a*x) * sin(a*x/2)^2 - a/2 * sin(a*x) * sin(n*a*x/2)^2) / sin(a*x/2)^4 )
equationN = simple( subs(equation,n,3) );
solve(equationN,x)
ans =
0
(2*pi)/(3*a)
-(2*pi)/(3*a)
and so on if you replace n you'll get multiple solutions. It's just a work-around but maybe it'll help?
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Giovanni
2011년 7월 27일
That's what solve outputs... I didn't look at the details of your equation so I am not sure how to answer that, but I would expect the solutions to have some periodicity, right?
Walter Roberson
2011년 7월 27일
2*RootOf(tan(Z)*n-tan(n*Z),Z)/a
is the general form.
For n=3, the general solutions are
(set of 4*Pi*Z/a) union set of (2*(Pi+2*Pi*Z)/a)
where Z ranges over all of the integers.
The solutions for higher n are more extensive -- e.g., 10 sets for n=6 .
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Stefan Wehmeier
2011년 7월 28일
There is no closed-form symbolic solution for general n, and for given n nobody can say which solutions are between -5 and 5 without knowing a. If you insert integer values for n, the solution will be found, but the complete (infinite) solution can only be represented in the symbolic engine, not on the matlab level, so you may want to proceed with
evalin(symengine, 'solve(your_equation, x)')
or use feval.
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