Reduction of the mathematical model of some problems of mathematical economics to systems of differential equations that can be solved in quadratures

The article discusses the problems associated with the Ramsey — Kass — Koopmans mathematical model of economic growth. An auxiliary system of differential equations is being constructed, for which it is possible to obtain a solution in quadratures. Based on the obtained solution, the upper estimates of the consumption function are found. Using the upper estimates of the consumption function, we find the maximum value of the time interval in which there are solutions to the auxiliary system of differential equations for the considered parameter values.
Under a special initial condition, we show that there is a solution to the Cauchy problem $(K(t)$, $C(t))$ on the entire ray $t\in [0;+\infty)$ and both components increase and tend to the values we found.