It is given an ordinary differential equation $\ddot{x}(t;A)=A(p(t)\dot{x}(t;A)+q(t)x(t;A))$ with real parameter $A$, initial conditions $x(t=0;A)=1,\dot{x}(t=0;A)=0$ for all real $A$. If $r(A)$ is the root of $x(t;A)=0$, i.e. $x(r(A);A)=0$, is it possible to determine $r(A)$ (or at least determine some properties of it) in terms of the forms of $p(t),q(t)$? Especially important is the correlation between $r(A)$ and $A$, e.g. whether they are correlated linearly, exponentially, etc, given the forms of $p(t)$ and $q(t)$.
The $x(t;A)$ that I am dealing in my actual problem only has one root, so whenever necessary, this can be taken as an assumption. If the general case is unsolvable, then it is also good if the problem can be solved for special forms of $p(t),q(t)$, e.g. $p(ct)=c^\gamma p(t), q(ct)=c^\gamma q(t)$ for some real/complex $\gamma$.