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1. First-Order Ordinary Differential Equations

  1. _y_′ =f(y). Autonomous equation.
  2. _y_′ =f(x)g(y). Separable equation.
  3. g(x)_y_′ =_f_1(x)y +_f_0(x). Linear equation.
  4. g(x)_y_′ =_f_1(x)y +f n(x)y n. Bernoulli equation.
  5. _y_′ =f(y/x). Homogeneous equation
  6. _y_′ = _ay_2 +bx n. Special Riccati equation.
  7. _y_′ = _y_2 +f(x)y −_a_2 − af(x). Riccati equation, special case 1.
  8. _y_′ =f(x)y_2 +ay − ab −_b_2_f(x). Riccati equation, special case 2.
  9. _y_′ = _y_2 +xf(x)y +f(x). Riccati equation, special case 3.
  10. _y_′ =f(x)y_2 −_ax n f(x)y + anx _n_−1. Riccati equation, special case 4.
  11. _y_′ =f(x)_y_2 +anx n_−1 −_a_2_x_2_n f(x). Riccati equation, special case 5.
  12. y_′ = −(n + 1)x n y_2 +x n+1_f(x)y −_f(x). Riccati equation, special case 6.
  13. x _y_′ =f(x)_y_2 +ny + ax_2_n f(x). Riccati equation, special case 7.
  14. x _y_′ =x_2_n f(x)y_2 + [ax n f(x) −_n]y +bf(x). Riccati equation, special case 8.
  15. y_′ =f(x)y_2 +g(x)y −_a_2_f(x) −_ag(x). Riccati equation, special case 9.
  16. _y_′ =f(x)y_2 +g(x)y +anx n_−1 −_a_2_x_2_n f(x) −_ax n g(x). Riccati equation, special case 10.
  17. _y_′ =ae λx _y_2 +ae λx f(x)y +λ f(x). Riccati equation, special case 11.
  18. _y_′ =f(x)y_2 −_ae λx f(x)y +aλe λx. Riccati equation, special case 12.
  19. _y_′ =f(x)y_2 +aλe λx −_a_2_e_2_λx f(x). Riccati equation, special case 13.
  20. _y_′ =f(x)_y_2 +λy + ae_2_λx f(x). Riccati equation, special case 14.
  21. _y_′ =_y_2 −_f_2(x) +_f_′(x). Riccati equation, special case 15.
  22. _y_′ =f(x)y_2 −_f(x)g(x)y +_g_′(x). Riccati equation, special case 16.
  23. _y_′ =f(x)_y_2 +g(x)y +h(x). General Riccati equation.
  24. _yy_′ =y + f(x). Abel equation of the second kind in the canonical form.
  25. _yy_′ =f(x)y + g(x). Abel equation of the second kind.
  26. _yy_′ =f(x)_y_2 +g(x)y +h(x). Abel equation of the second kind.
  27. _y_′ =f(ax + by + c).
  28. y_′ =f(y +ax n + b) −_anx _n_−1.
  29. _y_′ = (y/x)f(x n y m). Generalized homogeneous equation.
  30. _y_′ = −(n/m)(y/x) +y k f(x)g(x n y m).
  31. _y_′ =f((ax +by + c)/(αx + βy +γ)).
  32. _y_′ =x n_−1_y_1−_m f(ax n +by m).
  33. [x n f(y) +xg(y)]_y_′ =h(y).
  34. x[f(x n y m) +m x k g(x n y m)]y_′ =y[h(x n y m) −_n x k g(x n y m)].
  35. x[f(x n y m) +m y k g(x n y m)]y_′ =y[h(x n y m) −_n y k g(x n y m)].
  36. x[sf(x n y m) −mg(x k y s)]y_′ =y[ng(x k y s) −_kf(x n y m)].
  37. [f(y) +amx n y _m_−1]_y_′ +g(x) +anx n_−1_y m = 0.
  38. _y_′ =e_−_λx f(e λx y).
  39. _y_′ =e λy f(e λy x).
  40. _y_′ =yf(e αx y m).
  41. _y_′ =x_−1_f(x n e αy).
  42. _y_′ =f(x)e λy +g(x).
  43. _y_′ = −_nx_−1 +f(x)g(x n e y).
  44. _y_′ = −(α/m)y +y k f(x)g(e αx y m).
  45. _y_′ =e αx_−_βy f(ae αx +be βy).
  46. [e αx f(y) +aβ]_y_′ +e βy g(x) + aα = 0.
  47. x[f(x n e αy) +αyg(x n e αy)]y_′ =h(x n e αy) −_nyg(x n e αy).
  48. [f(e αx y m) +mxg(e αx y m)]y_′ =y[h(e αx y m) −_αxg(e αx y m)].

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