Exact Solutions > Systems of Ordinary Differential Equations > Linear Systems of Two Ordinary Differential Equations

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1. Linear Systems of Two Ordinary Differential Equations

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

1. x′ = ax + by,   y′ = cx + dy.
2. x′ = a₁x + b₁y + c₁,   y′ = a₂x + b₂y + c₂.
3. x′ = f(t)x + g(t)y,   y′ = g(t)x + f(t)y.
4. x′ = f(t)x + g(t)y,   y′ = −g(t)x + f(t)y.
5. x′ = f(t)x + g(t)y,   y′ = ag(t)x + [f(t) + bg(t)]y.
6. x′ = f(t)x + g(t)y,   y′ = a[f(t) + ah(t)]x + a[g(t) − h(t)]y.
7. x′ = f(t)x + g(t)y,   y′ = h(t)x + p(t)y.

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

8. x″ = ax + by,   y″ = cx + dy.
9. x″ = a₁x + b₁y + c₁,   y″ = a₂x + b₂y + c₂.
10. x″ − ay′ + bx = 0,   y″ + ax′ + by = 0.
11. x″ + a₁x′ + b₁y′ + c₁x + d₁y = k₁e^iωt,   y″ + a₂x′ + b₂y′ + c₂x + d₂y = k₂e^iωt.
12. x″ = a(ty′ − y),   y″ = b(tx′ − x).
13. x″ = f(t)(a₁x + b₁y),   y″ = f(t)(a₂x + b₂y).
14. x″ = f(t)(a₁x′ + b₁y′),   y″ = f(t)(a₂x′ + b₂y′).
15. x″ = af(t)(ty′ − y),   y″ = bf(t)(tx′ − x).
16. t²x″ + a₁tx′ + b₁ty′ + c₁x + d₁y = 0,   t²y″ + a₂tx′ + b₂ty′ + c₂x + d₂y = 0.
17. (αt² + βt + σ)²x″ = ax + by,   (αt² + βt + σ)²y″ = cx + dy.
18. x″ = f(t)(tx′ − x) + g(t)(ty′ − y),   y″ = h(t)(tx′ − x) + p(t)(ty′ − y).