Systems of linear ordinary differential equations

Homogeneous systems of linear ODEs with constant coefficients

Let \(\mathbb{K} = \mathbb{R} \lor \mathbb{C}\), \(n \in \mathbb{N}\) and \(A \in \mathbb{R}^{n \times n}\). Seek differentiable functions \(y:\mathbb{R} \to \mathbb{K}^n\) such that

\[ \mathbf{\dot y}(t) = A \mathbf{y}(t), \qquad t \in \mathbb{R} \]

The solutions from a linear space, therefore the general solutions can be written as,

\[ \mathbf{y}(t) = \sum_{k=1}^n c_k \mathbf{y}_k(t), \qquad c_k \in \mathbb{K} \]

where \(\{\mathbf{y_1}, \dots, \mathbf{y_n}\}\) is a linear independent set of solutions, i.e. the basis of the solutions space.

Assume now that \(A\) is diagonalizable, and let \(\{\mathbf{v_1}, \dots, \mathbf{v_n}\}\) be a basis of \(\mathbb{K}^n\) consisting of eigenvectors of A.

\[ AV = VD, \qquad \text{with } D = \begin{pmatrix} \lambda_1 & & \\ & \ddots & \\ & & \lambda_n \end{pmatrix} \]

then \(A = VDV^{-1}\), let \(\mathbf{z}(t) = V^{-1} \mathbf{y}(t)\)

\[ \begin{array}{ll} &\mathbf{\dot z} = V^{-1} \mathbf{\dot y} = V^{-1} A \mathbf{y} = V^{-1} V D V^{-1} = D \mathbf{z}, \\ & \mathbf{\dot z} = D \mathbf{z} \implies \mathbf{z}(t) = \mathbf{c} e^{\lambda t}. \end{array} \]

Obtaining the general solution

\[\mathbf{y}(t) = V \mathbf{z}(t) = \sum_{k=1}^n c_k \mathbf{v_k} e^{\lambda_k t}. \]

Inhomogeneous systems of linear ODEs with constant coefficients

Let \(I \subseteq \mathbb{R}\) be an interval, \(\mathbf{f}: I \to \mathbb{R}\) continuous. Find functions \(\mathbf{y}: I \to \mathbb{R}^n\) such that

\[ \mathbf{\dot y}(t) = A \mathbf{y}(t) + \mathbf{f}(t), \qquad t \in I. \qquad (*) \]

Theorem: let \(\mathbf{y}_p: I \to \mathbb{R}^n\) a particular solution for \((*)\) and \(\mathbf{y}_h\) the general solution to the homegeneous system. Then the general solutions of the inhomogeneous system \((*)\) is given by

\[ \mathbf{y}(t) = \mathbf{y}_p(t) + \mathbf{y}_h(t), \qquad t \in I \]

Proof:

Similar to 1d case, will be added later.

Method of variation of parameters

Let \(\{\mathbf{y_1}, \dotsc, \mathbf{y_n}\}\) be a basis for the solution space of the homogeneous system. Ansatz:

\[ \mathbf{y}_p(t) = \sum_{k=1}^n c_k(t) \mathbf{y}_k(t) = (\mathbf{y}_1, \dots, \mathbf{y}_n) \begin{pmatrix} c_1(t) \\ \vdots \\ c_n(t) \end{pmatrix} = Y(t) \mathbf{c}(t), \]

where \(c_1(t), \dots, c_n(t): I \to \mathbb{R}\) are to be determined.

Then:

\[ \begin{align*} \mathbf{\dot y}_p &= \sum_{k=1}^n \dot c_k(t) \mathbf{y}_k(t) + \sum_{k=1}^n c_k(t) \mathbf{\dot y}_k(t), \\ &= \sum_{k=1}^n \dot c_k(t) \mathbf{y}_k(t) + A \sum_{k=1}^n c_k(t) \mathbf{y}_k(t), \\ &= Y(t) \mathbf{\dot c}(t) + A \mathbf{y}_p(t). \end{align*} \]

Demanding that: \(Y(t) \mathbf{\dot c}(t) = \mathbf{f}(t)\) is the Wronskian. Then \(\mathbf{\dot c}(t) = Y^{-1}(t) \mathbf{f}(t) \iff Y(t)\) is nonsingular. Then solve for \(\mathbf{c}(t)\).