Exact Solutions > Ordinary Differential Equations > First-Order Ordinary Differential Equations

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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′ = f₁(x)y + f₀(x).    Linear equation.
4. g(x)y′ = f₁(x)y + f_n(x)yⁿ.    Bernoulli equation.
5. y′ = f(y/x).    Homogeneous equation
6. y′ = ay² + bxⁿ.    Special Riccati equation.
7. y′ = y² + f(x)y − a² − af(x).    Riccati equation, special case 1.
8. y′ = f(x)y² + ay − ab − b²f(x).    Riccati equation, special case 2.
9. y′ = y² + xf(x)y + f(x).    Riccati equation, special case 3.
10. y′ = f(x)y² − axⁿf(x)y + anxⁿ⁻¹.    Riccati equation, special case 4.
11. y′ = f(x)y² + anxⁿ⁻¹ − a²x²ⁿf(x).    Riccati equation, special case 5.
12. y′ = −(n + 1)xⁿy² + xⁿ⁺¹f(x)y − f(x).    Riccati equation, special case 6.
13. xy′ = f(x)y² + ny + ax²ⁿf(x).    Riccati equation, special case 7.
14. xy′ = x²ⁿf(x)y² + [axⁿf(x) − n]y + bf(x).    Riccati equation, special case 8.
15. y′ = f(x)y² + g(x)y − a²f(x) − ag(x).    Riccati equation, special case 9.
16. y′ = f(x)y² + g(x)y + anxⁿ⁻¹ − a²x²ⁿf(x) − axⁿg(x).    Riccati equation, special case 10.
17. y′ = ae^λxy² + ae^λxf(x)y + λf(x).    Riccati equation, special case 11.
18. y′ = f(x)y² − ae^λxf(x)y + aλe^λx.    Riccati equation, special case 12.
19. y′ = f(x)y² + aλe^λx − a²e^2λxf(x).    Riccati equation, special case 13.
20. y′ = f(x)y² + λy + ae^2λxf(x).    Riccati equation, special case 14.
21. y′ = y² − f²(x) + f′(x).    Riccati equation, special case 15.
22. y′ = f(x)y² − f(x)g(x)y + g′(x).    Riccati equation, special case 16.
23. y′ = f(x)y² + 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)y² + g(x)y + h(x).    Abel equation of the second kind.
27. y′ = f(ax + by + c).
28. y′ = f(y + axⁿ + b) − anxⁿ⁻¹.
29. y′ = (y/x)f(xⁿy^m).    Generalized homogeneous equation.
30. y′ = −(n/m)(y/x) + y^kf(x)g(xⁿy^m).
31. y′ = f((ax + by + c)/(αx + βy + γ)).
32. y′ = xⁿ⁻¹y^1−mf(axⁿ + by^m).
33. [xⁿf(y) + xg(y)]y′ = h(y).
34. x[f(xⁿy^m) + mx^kg(xⁿy^m)]y′ = y[h(xⁿy^m) − nx^kg(xⁿy^m)].
35. x[f(xⁿy^m) + my^kg(xⁿy^m)]y′ = y[h(xⁿy^m) − ny^kg(xⁿy^m)].
36. x[sf(xⁿy^m) − mg(x^ky^s)]y′ = y[ng(x^ky^s) − kf(xⁿy^m)].
37. [f(y) + amxⁿy^m−1]y′ + g(x) + anxⁿ⁻¹y^m = 0.
38. y′ = e^−λxf(e^λxy).
39. y′ = e^λyf(e^λyx).
40. y′ = yf(e^αxy^m).
41. y′ = x⁻¹f(xⁿe^αy).
42. y′ = f(x)e^λy + g(x).
43. y′ = −nx⁻¹ + f(x)g(xⁿe^y).
44. y′ = −(α/m)y + y^kf(x)g(e^αxy^m).
45. y′ = e^αx−βyf(ae^αx + be^βy).
46. [e^αxf(y) + aβ]y′ + e^βyg(x) + aα = 0.
47. x[f(xⁿe^αy) + αyg(xⁿe^αy)]y′ = h(xⁿe^αy) − nyg(xⁿe^αy).
48. [f(e^αxy^m) + mxg(e^αxy^m)]y′ = y[h(e^αxy^m) − αxg(e^αxy^m)].