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Exact Solutions > First-Order Partial Differential Equations > Quasilinear Partial Differential Equations

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2. First-Order Quasilinear Partial Differential Equations

2.1. Equations of the Form f(x, y)wx + g(x, y)wy = h(x, y, w)

  1. wx + awy = f(x)w + g(x)wk.
  2. wx + awy = f(x) + g(x)eλw.
  3. awx + bwy = f(w).
  4. awx + bwy = f(x)g(w).
  5. wx + awy = f(x)g(y)h(w).
  6. axwx + bywy = f(w).
  7. aywx + bxwy = f(w).
  8. axnwx + bykwy = f(w).
  9. aynwx + bxkwy = f(w).
  10. aeλxwx + beβywy = f(w).
  11. aeλywx + beβxwy = f(w).
  12. f(x)wx + g(y)wy = h(w).
  13. f(y)wx + g(x)wy = h(w).

2.2. Equations of the Form wx + f(x, y, w)wy = 0

  1. wx + [aw + yf(x)]wy = 0.
  2. wx + [aw + f(y)]wy = 0.
  3. wx + f(w)wy = 0.
  4. wx + [f(w) + ax]wy = 0.
  5. wx + [f(w) + ay]wy = 0.
  6. wx + [f(w) + g(x)]wy = 0.
  7. wx + [f(w) + g(y)]wy = 0.
  8. wx + [yf(w) + g(x)]wy = 0.
  9. wx + [xf(w) + yg(w) + h(w)]wy = 0.
  10. wx + f(x)g(y)h(w)wy = 0.

2.3. Equations of the Form wx + f(x, y, w)wy = g(x, y, w)

  1. wx + awwy = f(x).
  2. wx + awwy = f(y).
  3. wx + [aw + f(x)]wy = g(x).
  4. wx + f(w)wy = g(x).
  5. wx + f(w)wy = g(y).
  6. wx + f(w)wy = g(w).
  7. wx + [f(w) + g(x)]wy = h(x).
  8. wx + [f(w) + g(x)]wy = h(w).
  9. wx + [f(w) + yg(x)]wy = h(x).
  10. wx + f(x, w)wy = g(x).
  11. wx + f(x, w)wy = g(w).

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