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.