MathLabs

Problem 4

Find all real solutions x1,…,x5x_1,\ldots,x_5 of xi+xi+2=yxi+1x_i+x_{i+2}=yx_{i+1} for i=1,…,5i=1,\ldots,5, where subscripts are reduced modulo 55.
Step 2 of 4: Separate the value y=2y=2
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In plain words

The sum detects the uniform eigenmode of the five-cycle.

(y−2)(x1+x2+x3+x4+x5)=0(y-2)(x_1+x_2+x_3+x_4+x_5)=0
Detailed analysis

Summing all five equations gives 2∑xi=y∑xi2\sum x_i=y\sum x_i. If y=2y=2, the constant sequences x1=⋯=x5=sx_1=\cdots=x_5=s are solutions. The zero sequence is included for every yy.