8.1.4.3 Geometric and physical applications

Problem: A material point of mass \( m \) moves linearly under the action of a force of attraction to a fixed center, proportional to the distance from the point to the center (the proportionality coefficient \( k>0 \) ). The drag force of the environment is proportional to the velocity (the proportionality coefficient \( \lambda>0) \). At the initial moment, the distance from the point to the center is equal to \( a \), and the velocity is directed along the straight line connecting the point with the center, and equal to \( v_{0} \). Find the law of motion of a point provided that \( \lambda^{2}<4 m k \).