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3. First-Order Nonlinear Partial Differential Equations

Preliminary remarks. For first-order partial differential equations in two independent variables, an exact solution

(*)    w = Φ(x, y, C1, C2)

that depends on two arbitrary constants C1 and C2 is called a complete integral. The general integral (general solution) can be represented in parametric form by using the complete integral (*) and the two equations

C2 = f(C1),

ΦC1 + ΦC2f′(C1) = 0,

where f(C1) is an arbitrary function, the prime stands for the derivative, and ΦC1 and ΦC2 are partial derivatives.

References

  1. E. Kamke, Differentialgleichungen: Losungsmethoden und Losungen, II, Partielle Differentialgleichungen Erster Ordnung fur eine gesuchte Funktion, Akad. Verlagsgesellschaft Geest & Portig, Leipzig, 1965.
  2. A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002.

3.1. Equations Quadratic in One Derivative

  1. wx + a(wy)2 = by.
  2. wx + a(wy)2 + by2 = 0.
  3. wx + a(wy)2 = f(x) + g(y).
  4. wx + a(wy)2 = f(x)y + g(x).
  5. wx + a(wy)2 = f(x)w + g(x).
  6. wx − f(w)(wy)2 = 0.
  7. f1(x)wx + f2(y)(wy)2 = g1(x) + g2(y).
  8. wx + a(wy)2 + bwy = f(x) + g(y).
  9. wx + a(wy)2 + bwy = f(x)y + g(x).
  10. wx + a(wy)2 + bwy = f(x)w + g(x).

3.2. Equations Quadratic in Two Derivatives

  1. a(wx)2 + b(wy)2 = c.    Differential equation of light rays (for a = b).
  2. (wx)2 + (wy)2 = a − 2by.
  3. (wx)2 + (wy)2 = a(x2 + y2)−1/2 + b.
  4. (wx)2 + (wy)2 = f(x).
  5. (wx)2 + (wy)2 = f(x) + g(y).
  6. (wx)2 + (wy)2 = f(x2 + y2).
  7. (wx)2 + (wy)2 = f(w).
  8. (wx)2 + x−2(wy)2 = f(x).
  9. (wx)2 + f(x)(wy)2 = g(x).
  10. (wx)2 + f(y)(wy)2 = g(y).
  11. (wx)2 + f(w)(wy)2 = g(w).
  12. f1(x)(wx)2 + f2(y)(wy)2 = g1(x) + g2(y).

3.3. Equations with Arbitrary Nonlinearities in Derivatives

  1. wx + f(wy) = 0.
  2. wx + f(wy) = g(x).
  3. wx + f(wy) = g(x)y + h(x).
  4. wx + f(wy) = g(x)w + h(x).
  5. wx − F(x, wy) = 0.
  6. wx + F(x, wy) = aw.
  7. wx + F(x, wy) = g(x)w.
  8. F(wx, wy) = 0.
  9. w = xwx + ywy + F(wx, wy).    Clairaut's equation.
  10. F1(x, wx) = F2(y, wy).    Separable equation.
  11. F1(x, wx) + F2(y, wy) + aw = 0.    Separable equation.
  12. F1(x, wx/w) + wkF2(y, wy/w) = 0.
  13. F1(x, wx) + eλwF2(y, wy) = 0.
  14. F1(x, wx/w) + F2(y, wy/w) = k ln w.
  15. wx + yF1(x, wy) + F2(x, wy) = 0.
  16. F(wx + ay, wy + ax) = 0.
  17. (wx)2 + (wy)2 = F(x2 + y2, ywx − xwy).
  18. F(x, wx, wy) = 0.
  19. F(ax + by, wx, wy) = 0.
  20. F(w, wx, wy) = 0.
  21. F(ax + by + cw, wx, wy) = 0.
  22. F(x, wx, wy, w − ywy) = 0.
  23. F(w, wx, wy, xwx + ywy) = 0.
  24. F(ax + by, wx, wy, w − xwx − ywy) = 0.
  25. F(x, wx, G(y, wy)) = 0.    Separable equation.

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