I have an object with initial displacement $\vec{s_0}$ and initial velocity $\vec{v_0}$.
This object is accelerated as: $$\vec{a}=-\frac{v^2}{s^2}\vec{s}$$ where $v$ and $s$ are the sizes of velocity $\vec{v}$ and displacement $\vec{s}$ respectively. This comes from the formula of centrifugal acceleration. I want to find the general solution to this.
I attempted: Let $$\vec{s}=x\hat{i}+y\hat{j}+z\hat{k};\quad\vec{v}=\dot{x}\hat{i}+\dot{y}\hat{j}+\dot{z}\hat{k};\quad\vec{a}=\ddot{x}\hat{i}+\ddot{y}\hat{j}+\ddot{z}\hat{k}$$
Differentiating with respect to the time $t$. This results in:
$$\ddot{x}=-\frac{\dot{x}^2+\dot{y}^2+\dot{z}^2}{x^2+y^2+z^2}\;x$$ $$\ddot{y}=-\frac{\dot{x}^2+\dot{y}^2+\dot{z}^2}{x^2+y^2+z^2}\;y$$ $$\ddot{z}=-\frac{\dot{x}^2+\dot{y}^2+\dot{z}^2}{x^2+y^2+z^2}\;z$$
And I have no idea how to move on. Maybe I shouldn't use Cartesian..?