Let us consider the linear combination ψ(x)=C1eikx+C2e−ikx, where C1,C2 - constants. Then,
dx2d2ψ(x)=C1(ik)2eikx+C2(−ik)2e−ikx=−k2(C1eikx+C2e−ikx)=−k2ψ(x).
Hence, ψ(x) is an eigenfunction of differential operator dx2d2 with corresponding eigenvalue −k2.
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