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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′ = eλxF(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)

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