Hint: you might solve the problem in plane polar coordinates, since your monopole will never leave the plane perpendicular to the current-carrying wire. The magnetic field induced by the wire will vary as b/r, so the equations of motion will be $$ \ddot{\vec r}=\hat{r} (\ddot r -r\dot\theta^2)+ \hat\theta (r\ddot\theta+2\dot r \dot \theta)= {b\over r} $$ so that $$ \ddot r = r\dot\theta ^2, ~~~~ r\ddot\theta+2\dot r \dot \theta={b\over r}~~. $$ You may solve these and perhaps eliminate t from them, so as to express your parametric spiral, $r(\theta)$ .

Hint: you might solve the problem in plane polar coordinates, since your monopole will never leave the plane perpendicular to the current-carrying wire. The magnetic field induced by the wire will vary as b/r, so the equations of motion will be $$ \ddot{\vec r}=\hat{r} (\ddot r -r\dot\theta^2)+ \hat\theta (r\ddot\theta+2\dot r \dot \theta)= \hat \theta {b\over r} $$ so that $$ \ddot r = r\dot\theta ^2, ~~~~ r\ddot\theta+2\dot r \dot \theta={b\over r}~~. $$ You may solve these and perhaps eliminate t from them, so as to express your parametric spiral, $r(\theta)$ .