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Can the general solution of a linear ordinary differential equation be expressed in terms of its initial conditions?

It seems that I have seem this kind of representation.

It makes "some sense" to me but I hope to know if there is some "proof" or explanation of why it can be?

To be specific, for a n-th order ODE,

the solution is something like

y = y(xo)(something) + y'(xo)(something) + ... + y(n-1)(xo)(something)...

why?