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3. Nonlinear Systems of Two Ordinary Differential Equations

3.1. Systems of First-Order Ordinary Differential Equations; x = x(t), y = y(t)

1. x′ = xⁿF(x, y),   y′ = g(y) F(x, y).
2. x′ = e^λxF(x, y),   y′ = g(y) F(x, y).
3. x′ = F(x, y),   y′ = G(x, y).    Autonomous system of general form.
4. x′ = f₁(x) g₁(y) Φ(x, y, t),   y′ = f₂(x) g₂(y) Φ(x, y, t).
5. x = tx′ + F(x′, y′),   y = ty′ + G(x′, y′).    Clairaut system.

3.2. Systems of Second-Order Ordinary Differential Equations; x = x(t), y = y(t)

6. x″ = xf(ax − by) + g(ax − by),   y″ = yf(ax − by) + h(ax − by).
7. x″ = xf(y/x),   y″ = yg(y/x).
8. x″ = kxr⁻³,   y″ = kyr⁻³;    r = (x² + y²)^1/2.    Equations of motion of a point mass in gravitational field.
9. x″ = xf(r),   y″ = yf(r);    r = (x² + y²)^1/2.    Equations of motion of a point mass in central force field.
10. x″ = xf(x² + y², y/x) − yg(y/x),   y″ = yf(x² + y², y/x) + xg(y/x).
11. x″ = −f(y)g(v)x′,   y″ = −f(y)g(v)y′ − a;    v = [(x′)² + (y′)²]^1/2.    Equations of motion of a projectile.
12. x″ + a(t)x = x⁻³f(y/x),   y″ + a(t)y = y⁻³g(y/x).    Generalized Ermakov (Yermakov) system.
13. x″ = x⁻³F(x/φ(t), y/φ(t)),   y″ = y⁻³G(x/φ(t), y/φ(t));    φ(t) = (at² + bt + c)^1/2.
14. x″ = f(y′/x′),   y″ = g(y′/x′).
15. x″ = xΦ(x, y, t, x′, y′),   y″ = yΦ(x, y, t, x′, y′).
16. x″ + x⁻³f(y/x) = xΦ(x, y, t, x′, y′),   y″ + y⁻³g(y/x) = yΦ(x, y, t, x′, y′).
17. x″ = F(t, tx′ − x, ty′ − y),   y″ = G(t, tx′ − x, ty′ − y).
18. x″ = x′Φ(x, y, t, x′, y′) + f(y),   y″ = −y′Φ(x, y, t, x′, y′) + g(x).
19. x″ = ay′Φ(x, y, t, x′, y′) + f(x),   y″ = bx′Φ(x, y, t, x′, y′) + g(y).
20. x″ = f(y′)Φ(x, y, t, x′, y′),   y″ = g(x′)Φ(x, y, t, x′, y′).