8.1.4.2 Geometric and physical applications

Problem: A material point of mass \( m \) moves linearly under the influence of repulsive force from 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 point is at a distance \( a \) from the center, the velocity is equal to \( v_{0} \) is directed along the straight line, connecting the point with the center. Find the law of motion of the point.