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In an earlier version 8 (ParametricNDSolve not available) I tried to integrate with different initial values $ (yi= 0.1, 0.4,0.8); (zi^2+yi^2=1) $ radially on unit semi circle to sketch tractrices but not successful. Is there a workaround?

$$ \sin \phi = y/a,\, \tan \phi= \frac{dy}{dz} $$

```
a = 1; zmax = 1; ri = 0.8; zi = -Sqrt[a^2 - ri^2]; ar = 0.5;
{zi, ri}
NDSolve[{R'[z]/(R[z] (1 + R'[z]^2)^0.5) == 1/a, R[0] == ri},
R, {z, zi, zmax}];
r[t_] = R[t] /. First[%];
ps8 = Plot[{r[z], 0}, {z, zi, zmax}, PlotStyle -> {Red, Thick},
AspectRatio -> ar, GridLines -> Automatic];
semicirc =
ParametricPlot[a {Cos[t], Sin[t]}, {t, 0, Pi}, AxesOrigin -> {0, 0},
GridLines -> Automatic];
Show[{semicirc, ps1, ps4, ps8}, PlotRange -> All]
```

Thanks in advance.

What is your version? – Alex Trounev – 2019-03-30T22:32:13.573

Mathematica Version 8 – Narasimham – 2019-03-30T22:44:45.587