I am having some trouble regarding this second order homogeneous differential equation, the one which appears from the title. I am supposed to finde the complete solution of (1)

(1) y''+k2y=0, where k>0 is a constant

I am aware of the fact that if i use the Wronskian-determinant i get the complete solution with the arbitary constants c1 and c2

y=c1sinkx+c2coskx

as i have W(sin(kx), cos(kx)) =-ksin2kx-kcos2kx=-k≠0

but where the hell do sinkx and coskx come from? I'm so damn confused, and very pissed!

My book claims that by executing a sample we can conclude that (1) has the solutions coskx and sinkx, where do these come from and what is up with all this sample thing?