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$x(t)=a\cdot \sin(t\sqrt{k/m})$ is the only solution that satisfies $x(0)=0$. If you then impose $x(c)=0$, you only get solutions for certain values of $k$. These values are the eigenvalues of the differential operator $-m d^2/d^2 t$ acting on the vector space of functions satisfying the boundary conditions. I'm not sure what you're looking for in terms of physical meaning. This just means that you can only get solutions subject to your given conditions for certain values of $k$. If you set up a harmonic oscillator, no matter what you do, you cannot satisfy your boundary conditions unless the spring constant has one of these special values.