Hi, please review my answer, I suspect I am missing some fine points...
y(x) is a solution to a 2nd order, linear, homogeneous ODE. Also y(x0)=y0 and dy/dz=y'0
Show that y(x) is unique, in that no other solution passes through (x0, y0) with a slope of y'0.
Expanding y(x) in a Taylor series, $...