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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 $\omega(t)$ isa non-zero arbitrary function and the connection coefficients $\Gamma^{a}_{bc}$are computed from the kinetic metric (kinetic energy) of the system. In orderto 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 unknowncoefficients $K_{ab}, K_{a}, K$ are tensors depending on $t, q^{a}$ and imposethe condition $\frac{dI}{dt}=0$. This condition leads to a system of partialdifferential 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 aKilling 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 metricassuming the coefficients to be functions of $t$. In both cases, this leads toa 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 QFIswhich we collect in a Theorem. Next, we specify the quantities $Q^{a}(q)$ to bethe generalized time-dependent Kepler potential $V=-\frac{\omega (t)}{r^{\nu}}$and determine the functions $\omega(t)$ for which QFIs are admitted. We extendthe discussion to the non-linear differential equation$\ddot{x}=-\omega(t)x^{\mu }+\phi (t)\dot{x}$ $(\mu \neq -1)$ and compute therelation between the coefficients $\omega(t), \phi(t)$ so that QFIs areadmitted. We apply the results to determine the QFIs of the generalizedLane-Emden equation.

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