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

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1. First-Order Linear Partial Differential Equations

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

  1. wx + [f(x)y + g(x)]wy = 0.
  2. wx + [f(x)y + g(x)yk]wy = 0.
  3. wx + [f(x)eλy + g(x)]wy = 0.
  4. f(x)wx + g(y)wy = 0.
  5. [f(y) + amxnym−1]wx − [g(x) + anxn−1ym]wy = 0.
  6. [eαxf(y) + cβ]wx − [eβyg(x) + cα]wy = 0.
  7. wx + f(ax + by + c)wy = 0.
  8. wx + f(y/x)wy = 0.
  9. xwx + yf(xnym)wy = 0.
  10. wx + yf(eαxym)wy = 0.
  11. xwx + f(xneαy)wy = 0.

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

  1. awx + bwy = f(x).
  2. wx + awy = f(x)yk.
  3. wx + awy = f(x)eλy.
  4. awx + bwy = f(x) + g(y).
  5. wx + awy = f(x)g(y).
  6. wx + awy = f(x, y).
  7. wx + [ay + f(x)]wy = g(x).
  8. wx + [ay + f(x)]wy = g(x)h(y).
  9. wx + [f(x)y + g(x)yk]wy = h(x).
  10. wx + [f(x) + g(x)eλy]wy = h(x).
  11. axwx + bywy = f(x, y).
  12. f(x)wx + g(y)wy = h1(x) + h2(y).
  13. f(x)wx + g(y)wy = h(x, y).
  14. f(y)wx + g(x)wy = h(x, y).

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

  1. awx + bwy = f(x)w.
  2. awx + bwy = f(x)w + g(x).
  3. awx + bwy = [f(x) + g(y)]w.
  4. wx + awy = f(x, y)w.
  5. wx + awy = f(x, y)w + g(x, y).
  6. axwx + bywy = f(x)w + g(x).
  7. axwx + bywy = f(x, y)w.
  8. xwx + aywy = f(x, y)w + g(x, y).
  9. f(x)wx + g(y)wy = [h1(x) + h2(y)]w.
  10. f1(x)wx + f2(y)wy = aw + g1(x) + g2(y).
  11. f(x)wx + g(y)wy = h(x, y)w + r(x, y).
  12. f(y)wx + g(x)wy = h(x, y)w + r(x, y).

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