Question

I am not very sure if this is a proper question, but I'm trying to investigate what the area of math can offer in researching the differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const or in other notation: $r'(\theta)^2+r(\theta)^2=\theta^2$, $r(\theta=0)=0$