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

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

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](https://doi.org/10.2172/4096514) 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 .