Correct answer: (C) λ=k_n², where k_n are the positive roots of k−tan(kπ)=0
Explanation
Set k=√λ>0. The first condition gives y=C sin(kx). The second is C[sin(kπ)−k cos(kπ)]=0. For C≠0, cos(kπ) cannot vanish, so tan(kπ)=k. Thus **C**; the zero root is excluded by λ>0.
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