First-Order Partial Differential Equations, Nonlinear (original) (raw)

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

(*) w = Φ(x, y, _C_1, _C_2)

that depends on two arbitrary constants _C_1 and_C_2is called a complete integral. The general integral (general solution) can be represented in parametric form by using the complete integral (*) and the two equations

_C_2 = f(_C_1),

Φ_C_1 + Φ_C_2_f_′(_C_1) = 0,

where f(_C_1) is an arbitrary function, the prime stands for the derivative, and Φ_C_1 and Φ_C_2are 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 + _by_2 = 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. _f_1(x)wx + _f_2(y)(wy)2= _g_1(x) + _g_2(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 − 2_by_.
  3. (wx)2 + (wy)2= a(_x_2 + _y_2)−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(_x_2 + _y_2).
  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. _f_1(x)(wx)2+ _f_2(y)(wy)2= _g_1(x) + _g_2(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. _F_1(x, wx) = _F_2(y, wy). Separable equation.
  11. _F_1(x, wx) + _F_2(y, wy) + aw = 0. Separable equation.
  12. _F_1(x, wx/w) + _wkF_2(y, wy/w) = 0.
  13. _F_1(x, wx) + e_λ_w _F_2(y, wy) = 0.
  14. _F_1(x, wx/w) + _F_2(y, wy/w) = k ln w.
  15. wx + _yF_1(x, wy) + _F_2(x, wy) = 0.
  16. F(wx + ay, wy + ax) = 0.
  17. (wx)2 + (wy)2= F(_x_2 + _y_2, 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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