A for what Hans suggested I hadn't gotten as far and didn't mention ...
definition and solutions of a system of first order equations. 31.1. the pair of equations:
dy1/dt = f1(y1,y2,...,yn,t) = f1(t)x+g1(t)y+h1(t)
dy2/dt = f2(y1,y2,...,yn,t) = f2(t)x+g2(t)y+h2(t)
...
where f1 and f2 are functions of x,y,t fefined on a common set S, is called a system of two first order equations, the sol'n will be two functions on common interval I contained in S satisfying both.
means you can solve
dx/dt=t/x^2 , dy/dt=y/t^2
does not mean you are encouraged to write (for the sake of a book chapter)
dy/dx=y+2y
d^2y/dx^2=2y+2Y
because that is
dy1/dt = f1(y1,t) = f1(t)y1+h1(t)
dy1/dt = f2(y1,t) = f2(t)y1+h2(t)
while the following may be an identity in x, it may be (is in most cases) inconsistent
x' = f1(x)
x' = f2(x)
also a note that in "beginner's ODE" it is spelled out that fn(t) has no exponent restriction and must be the independent variable t
that being said i see why Hans decided y' and y'' are likely not the same.
i question Hans a little as since you said it's not a copied equation but your own (observation): maybe Mathematica's answer is the answer you seek. but i have not bench checked it.
you said your studying celestial mechanics, perhaps diff equation course is prerequisite (if it is not then you shouldn't need the answer the question), perhaps you did take the course a while back and need help remembering?
it's a fairly complex topic so unless the answer is a single function (often not), describing the answer would be difficult unless you had taken the course. solution, unless otherwise stated, means a complete and exact function but the answer may be implicit or have hidden answers and know these is the beginning of the course.