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Jun 25th, 2019
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  1. NDSolve[
  2. D[f[x, y], x, x]^2 + D[g[x, y], x, x]^2 + D[f[x, y], x, y]^2 == 0 &&
  3. D[f[x, y], y, y]^2 + D[g[x, y], y, y]^2 + D[f[x, y], x, y]^2 == 0 &&
  4. f[x, 0] == 0 && g[x, 0] == 0 && 1 == (D[f[x, q], q] /. q -> 0) &&
  5. 1 == (D[g[x, q], q] /. q -> 0) && 1 == (D[f[x, 0], x]) && 1 == (D[g[x, 0], x])
  6. ,{f, g}, {x, 0, 1}, {y, 0, 1}]
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