 EqWorld The World of Mathematical Equations Exact Solutions > Ordinary Differential Equations > Second-Order Nonlinear Ordinary Differential Equations ## 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′′ = Axnym.    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′′ = xk−2f(x−ky).    Generalized homogeneous equation.
10. y′′ = yx−2f(xnym).    Generalized homogeneous equation.
11. y′′ = y−3f(y(ax2 + bx + c)−1/2).
12. y′′ = e−axf(eaxy).
13. y′′ = yf(eaxym).
14. y′′ = x−2f(xneay).
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′)

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