File:Method of characteristics solution surface.svg

$ y u_x-x u_y-e^u=0 $ that passes through the curve $\Gamma=\{(x, y, u)=(s, \sin s, 0): s \in \mathbb{R}\}$.

```mathematica lhs1 = Sin[x*Cos[1 - Exp[-u]] - y*Sin[1 - Exp[-u]]]; rhs1 = x*Sin[1 - Exp[-u]] + y*Cos[1 - Exp[-u]];

surf1 = ContourPlot3D[

  lhs1 - rhs1 == 0, {x, -2, 2}, {y, -2, 2}, {u, -1, 2}, Mesh -> None,
   ContourStyle -> Opacity[0.6], PlotPoints -> 30, MaxRecursion -> 2,
   BoxRatios -> {1, 1, 1}, AxesLabel -> {"x", "y", "u"}, 
  PerformanceGoal -> "Quality"];

x1[s_, t_] := s Cos[t] + Sin[s] Sin[t]; y1[s_, t_] := Sin[s] Cos[t] - s Sin[t]; u1[t_] := -Log[1 - t]; (*t<1*)

sSeeds1 = Subdivide[-2, 2, 20];

curves1 =

 Table[ParametricPlot3D[{x1[s, t], y1[s, t], u1[t]}, {t, -1, 1}, 
   PlotStyle -> {Thick, Black}, PlotPoints -> 60, 
   MaxRecursion -> 1], {s, sSeeds1}];

plot1 = Show[surf1, Evaluate[curves1], Lighting -> "Neutral",

  Background -> White];

plot1