MATH528

MATH528 Lesson09: ODE Systems

Definition. The implicit form of a first-order ODE system is

$𝑭 (t, 𝒚, 𝒚^{'}) = 𝟎, with 𝑭 : ℝ \times ℝ^{n} \times ℝ^{n} \to ℝ^{n}, 𝒚 : ℝ \to ℝ^{n} .$

Definition. The explicit form of a first-order ODE system is

$𝒚^{'} = 𝒇 (t, 𝒚), with 𝒇 : ℝ \times ℝ^{n} \to ℝ^{n}, 𝒚 : ℝ \to ℝ^{n} .$

Definition. A linear system of first-order ODEs has form

$𝒚^{'} = 𝑨 (t) 𝒚 + 𝒈 (t), 𝑨 : ℝ \to ℝ^{n \times n}, 𝒈 : ℝ \to ℝ^{n} .$

Remark. An $n^{th}$ -order ODE

u^{(n)} = g (t, u, u^{'}, \dots, u^{(n - 1)})

can be rewritten as a system of first-order ODEs

(\begin{array}{c} y_{1}^{'} \\ y_{2}^{'} \\ ⋮ \\ y_{n - 1}^{'} \\ y_{n}^{'} \end{array}) = 𝒚^{'} = 𝒇 (t, 𝒚) = (\begin{array}{c} u \\ y_{1}^{'} \\ ⋮ \\ y_{n - 2}^{'} \\ g (t, 𝒚) \end{array}) .

Definition. The initial value problem for a first-order ODE system is

$𝒚^{'} = 𝒇 (t, 𝒚), with 𝒇 : ℝ \times ℝ^{n} \to ℝ^{n}, 𝒚 : ℝ \to ℝ^{n}, 𝒚 (t_{0}) = 𝑲 \in ℝ^{n} .$