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Differential equation orthogonal trajectories

Posted 7 months ago
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I am unable to get real solutions for the following

DSolve[y'[x] ==[ y[x]/(x + Sqrt[x^2 + y[x]^2]), y[x], x, 
 GeneratedParameters -> c]

I get

{{y[x] -> -Sqrt[-E^c[1] (E^c[1] + 2 I x)]}, {y[x] -> 
   Sqrt[-E^c[1] (E^c[1] + 2 I x)]}, {y[
    x] -> -Sqrt[-E^c[1] (E^c[1] - 2 I x)]}, {y[x] -> 
   Sqrt[-E^c[1] (E^c[1] - 2 I x)]}}

Here are some formula manipulations that yield complex-free results:

sol = DSolve[{y'[x] == y[x]/(x + Sqrt[x^2 + y[x]^2]),
    y[0] == c}, y[x], x];
Simplify[sol, c > 1];
% // PowerExpand // TrigToExp;
Simplify[%, c > 1] // PowerExpand // ExpandAll // Simplify // Union

Check the results before you use them, because PowerExpand uses assumptions that may not be true. These may or may not additional solutions:

Simplify[sol, -1 < c < 1];
% // PowerExpand // TrigToExp;
Simplify[%, -1 < c < 1] // PowerExpand // ExpandAll // 
  Simplify // Union
Simplify[sol, c < -1];
% // PowerExpand // TrigToExp;
Simplify[%, -1 < c < 1] // PowerExpand // ExpandAll // 
  Simplify // Union
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