I'm interested in plane curves with curvature inversely proportional to distance from the axis:

$$\kappa(t) = \left(\frac{x'(t) y''(t) - y'(t)x''(t)}{(x'(t)^2 + y'(t)^2)^{3/2}} \right) = \frac{1}{a x}$$$$\kappa(t) = \left(\frac{x'(t) y''(t) - y'(t)x''(t)}{(x'(t)^2 + y'(t)^2)^{3/2}} \right) = \frac{1}{a x(t)}$$

where $a$ is a parameter. Is there an (uncommon) name for such a curve? Is there a solution (even in terms of special functions) for the location along a curve as a function of $t$? (For initial conditions one might take $x'(0) = 0, y(0) = 0$.)

Previous studies of this problem

This curve is interesting for physicists: it is the shape of a toroidal solenoid made of filamentary conductors which are constant tension. The problem was studied in the mid-70s; a report freely available here gives formulas (in terms of Bessel functions and Struve functions) for arc lengths and enclosed area of the curves, but not for the shape itself.

Attempts to solve

I tried using Mathematica to solve in both Cartesian and polar coordinates, but came up empty-handed.

$$k r=\frac{\left(z'(r)^2+1\right)^{3/2}}{z''(r)}$$

DSolve[{k r == (1 + z'[r]^2)^(3/2)/z''[r], z[a] == 0, z'[a] == 0}, z,
  r]

$$ r''(t)=\frac{ \left(-\left(r'(t)^2+r(t)^2\right)^{3/2}+2 r(t) \cos (t) r'(t)^2+r(t)^3 \cos (t)\right)}{\cos(t) r(t)^2}, r'(0)=0, r(0)=a $$

DSolve[{(r^\[Prime]\[Prime])[t] == (
   Sec[t] (Cos[t] r[t]^3 + 
      2 Cos[t] r[t] Derivative[1][r][t]^2 - (r[t]^2 + 
        Derivative[1][r][t]^2)^(3/2)))/r[t]^2, 
  Derivative[1][r][0] == 0, r[0] == a}, r, t]

Similar problems

Curves where $\kappa$ is proportional to $x$ are Euler elasticae. Curves where $\kappa$ is proportional to distance from the origin is studied in https://www.jstor.org/stable/2589616 .