# 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