0

I'd like to solve analytically the following equation, where $\alpha_i$ and $\beta_i$ have known values in $\mathbb{R}$: \begin{equation} \sum_{i\leqslant N} \alpha_i\,\cos(\beta_i\,t)=0 \end{equation}

Would there be an exact solution to this problem?

Mathematica only finds approximated solutions, which prevents me from calculating the null space of the matrix whose determinant is the left-hand side of the above equation.

anderstood
  • 3,504
  • Probably no analytical solution unless you have some specific $\alpha$s and $\beta$s. – Yulia V Oct 06 '14 at 19:43
  • @YuliaV That's my intuition too but do you have reasons do think so? – anderstood Oct 06 '14 at 19:45
  • 3
    intuition - it is like a polynom, but more general, and there are no general solutions for polynoms for the degree greater than 4. No proof, analytical solution is like guilt; you presume that there is none unless shown otherwise. – Yulia V Oct 06 '14 at 19:48
  • This is possibly only if your $\alpha_,\beta_i$ are "nice." Do you have exact expressions for them? – Alex R. Oct 06 '14 at 21:09
  • @AlexR. Unfortunately not: they correspond to physical properties (masses & stiffnesses). I'll have to cope with the fact that there is no general solution... – anderstood Oct 06 '14 at 21:12

0 Answers0