Is there a method to simplify a differential equation with two variables? Specifically I am looking for a way to relate x(t) and y(t) in the following equation and I have a strong urge to say x(t)=y(t) but obviously there are answers where x(t)$\neq$y(t) satisfying the condition. How would you suggest to approach this problem? Or is it impossible to relate x(t) and y(t) simpler than the given equation?
$$\frac{x''(t)}{x(t)} = \frac{y''(t)}{y(t)}$$
note: sorry for the tags, I don't know which category this DE falls into.
Given $y$, we can solve for $x$. Note that writing $f := \frac{y''}y$, the equation reads $$ x''(t) = f(t)x(t) $$ Now writet $x = ay$, this gives \begin{align*} x'' &= a''y + 2a'y' + ay''\\ fx &= fay\\ x'' = fx &\iff a''y + 2a'y' + ay'' = fay\\ &\iff a''y + 2a'y' + ay'' = ay''\\ &\iff a''y + 2a'y' = 0 \end{align*} Write $b := a'$, we have $$ \frac{b'}b = -2\frac{y'}y $$ or $$ \log b(t) = -2 \log y(t) + C $$ for some constant $c$, taking the exponential leaves us with $$ b = \frac C{y^2} $$ for (some other) constant $C$. As $a = b'$, we have $$ a(t) = C_1 + \int_{t_0}^t \frac{C}{y^2(t)}\, dt $$ or $$ x(t) = C_1y(t) + \int_{t_0}^t \frac{C}{y^2(t)}\, dt \cdot y(t) $$ So $x$ and $y$ are related in the above way.