How to solve this vector differential equation concerning circular motion?

Question

Let $O$ be a fixed point. A string with natural length $a$ and modulus of elasticity $\lambda$ is attached to $O$ and a particle $P$ with mass $m$, which is free to move. Let $\def\r{\mathbf r} \vec{OP}=\r$. Let $\def\i{\mathbf i}\def\k{\mathbf k} \i,\k$ be the unit vector to the right and upwards, respectively. We can derive the equation of motion $$ m\ddot \r=-mg\k-\frac{\lambda}{a}\r+\frac{\lambda\r}{|\r|}. $$ Question: can we find a general solution of this equation? I know how to find the general solution of $$ m\ddot \r=-mg\k-\frac{\lambda}{a}\r, $$ but the last term is an eyesore - it makes the whole thing not linear.