Exact Solutions > Ordinary Differential Equations > Second-Order Nonlinear Ordinary Differential Equations

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3. Second-Order Nonlinear Ordinary Differential Equations

3.1. Ordinary Differential Equations of the Form y′′ = f(x, y)

1. y′′ = f(y).    Autonomous equation.
2. y′′ = Axⁿy^m.    Emden--Fowler equation.
3. y′′ + f(x)y = ay⁻³.    Ermakov (Yermakov) equation.
4. y′′ = f(ay + bx + c).
5. y′′ = f(y + ax² + bx + c).
6. y′′ = x⁻¹f(yx⁻¹).    Homogeneous equation.
7. y′′ = x⁻³f(yx⁻¹).
8. y′′ = x^−3/2f(yx^−1/2).
9. y′′ = x^k−2f(x^−ky).    Generalized homogeneous equation.
10. y′′ = yx⁻²f(xⁿy^m).    Generalized homogeneous equation.
11. y′′ = y⁻³f(y(ax² + bx + c)^−1/2).
12. y′′ = e^−axf(e^axy).
13. y′′ = yf(e^axy^m).
14. y′′ = x⁻²f(xⁿe^ay).
15. y′′ = (ψ′′/ψ)y + ψ⁻³f(y/ψ),    ψ = ψ(x).

3.2. Ordinary Differential Equations of the Form f(x, y)y′′ = g(x, y, y′)

16. y′′ − y′ = f(y).    Autonomous equation.
17. y′′ + f(y)y′ + g(y) = 0.    Lienard equation.
18. y′′ + [ay + f(x)]y′ + f′(x)y = 0.
19. y′′ + [2ay + f(x)]y′ + af(x)y² = g(x).
20. y′′ = ay′ + e^2axf(y).
21. y′′ = f(y)y′.
22. y′′ = [e^αxf(y) + α]y′.
23. xy′′ = ny′ + x^{2n + 1}f(y).
24. xy′′ = f(y)y′.
25. xy′′ = [x^kf(y) + k − 1]y′.
26. x²y′′ + xy′ = f(y).
27. (ax² + b)y′′ + axy′ + f(y) = 0.
28. y′′ = f(y)y′ + g(x).
29. xy′′ + (n + 1)y′ = x^{n − 1}f(yxⁿ).
30. g(x)y′′ + 1/2 g′(x)y′ = f(y).
31. y′′ = −ay′ + e^axf(ye^ax).
32. xy′′ = f(xⁿe^ay)y′.
33. x²y′′ + xy′ = f(xⁿe^ay).
34. yy′′ + (y′)² + f(x)yy′ + g(x) = 0.
35. yy′′ − (y′)² + f(x)yy′ + g(x)y² = 0.
36. yy′′ − n(y′)² + f(x)y² + ay^{4n − 2} = 0.
37. yy′′ − n(y′)² + f(x)y² + g(x)y^{n + 1} = 0.
38. yy′′ + a(y′)² + f(x)yy′ + g(x)y² = 0.
39. yy′′ = f(x)(y′)².
40. y′′ − a(y′)² + f(x)e^ay + g(x) = 0.
41. y′′ − a(y′)² + be^4ay + f(x) = 0.
42. y′′ + a(y′)² − 1/2 y′ = e^xf(y).
43. y′′ + α(y′)² = [e^βxf(y) + β]y′.
44. y′′ + f(y)(y′)² + g(y) = 0.
45. y′′ + f(y)(y′)² − 1/2 y′ = e^xg(y).
46. y′′ = xf(y)(y′)³.
47. y′′ = f(y)(y′)² + g(x)y′.
48. y′′ = f(x)g(xy′ − y).
49. y′′ = yx⁻²f(xy′/y).
50. gy′′ + 1/2 g′y′ = f(y)h(y′g^1/2),   g = g(x).
51. y′′ = f((y′)² + ay).