Problem 4.37

A particle of mass \(m\) moves due to the force \(\mathbf{F}=-\alpha m \mathbf{r}\) where \(\alpha\) is a positive constant, \(\mathbf{r}\) is the radius vector of the particle relative to the origin of coordinates. Find the trajectory of its motion if at the initial moment \(\mathbf{r}=r_{0} \mathbf{i}\) and the velocity \(\mathbf{v}=v_{0} \mathbf{j},\) where \(\mathbf{i}\) and \(\mathbf{j}\) are the unit vectors of the \(x\) and \(y\) axes.