Quadratic first integrals of time-dependent dynamical systems of the form $ddot{q}^{a}= -Gamma^{a}_{bc}dot{q}^{b} dot{q}^{c} -omega(t)Q^{a}(q)$

We consider the time-dependent dynamical system $ddot{q}^{a}= -Gamma_{bc}^{a}dot{q}^{b}dot{q}^{c}-omega(t)Q^{a}(q)$ where $omega(t)$ is a non-zero arbitrary function and the connection coefficients $Gamma^{a}_{bc}$ are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) $I$ we assume that $I=K_{ab}dot{q}^{a} dot{q}^{b} +K_{a}dot{q}^{a}+K$ where the unknown coefficients $K_{ab}, K_{a}, K$ are tensors depending on $t, q^{a}$ and impose the condition $frac{dI}{dt}=0$. This condition leads to a system of partial differential equations (PDEs) involving the quantities $K_{ab}, K_{a}, K,$ $omega(t)$ and $Q^{a}(q)$. From these PDEs, it follows that $K_{ab}$ is a Killing tensor (KT) of the kinetic metric. We use the KT $K_{ab}$ in two ways: a. We assume a general polynomial form in $t$ both for $K_{ab}$ and $K_{a}$; b. We express $K_{ab}$ in a basis of the KTs of order 2 of the kinetic metric assuming the coefficients to be functions of $t$. In both cases, this leads to a new system of PDEs whose solution requires that we specify either $omega(t)$ or $Q^{a}(q)$. We consider first that $omega(t)$ is a general polynomial in $t$ and find that in this case the dynamical system admits two independent QFIs which we collect in a Theorem. Next, we specify the quantities $Q^{a}(q)$ to be the generalized time-dependent Kepler potential $V=-frac{omega (t)}{r^{ u}}$ and determine the functions $omega(t)$ for which QFIs are admitted. We extend the discussion to the non-linear differential equation $ddot{x}=-omega(t)x^{mu }+phi (t)dot{x}$ $(mu eq -1)$ and compute the relation between the coefficients $omega(t), phi(t)$ so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane-Emden equation.