Application: Differential Equations

Section 8.4 Application: Differential Equations

In this section, we only focus on systems of first-order linear differential equations with constant coefficients. Such a system can be written in the form

\begin{align*} x_1'(t) \amp = a_{11}x_1(t)+a_{12}x_2(t) \\ x_2'(t) \amp = a_{21}x_1(t)+a_{22}x_2(t) \end{align*}

and the matrix \(A = \begin{pmatrix} a_{11} \amp a_{12} \\ a_{21} \amp a_{22} \end{pmatrix}\) is not diagonalizable.

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Theorem 8.4.1. Non-Diagonalizable 2×2 Matrices.

Suppose that an \(2 \times 2\) matrix \(A\) has an eigenvalue \(\lambda\in \mathbb{R}\) and \(A\) is not diagonalizable. Then the Jordan canonical form of \(A\) is

\begin{equation*} J = \begin{pmatrix} \lambda \amp 1 \\ 0 \amp \lambda \end{pmatrix} \end{equation*}

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Theorem 8.4.2. Solution of x’(t) = Ax(t) for Non-Diagonalizable 2×2 Matrix.

Let \(A\) be a \(2 \times 2\) matrix that is not diagonalizable with eigenvalue \(\lambda\) (of algebraic multiplicity 2), and Let \(P\) be an invertible matrix such that \(P^{-1}AP = \begin{pmatrix} \lambda \amp 1 \\ 0 \amp \lambda \end{pmatrix}\text{.}\)

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A fundamental set of solutions is:

\begin{equation*} \mathbf{x}_1(t) = e^{\lambda t}P\mathbf{e}_1, \quad \mathbf{x}_2(t) = e^{\lambda t}(tP\mathbf{e}_1 + P\mathbf{e}_2) \end{equation*}

where \(\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\text{.}\)

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Activity 8.4.1.

Find a fundamental set of solutions of

\begin{align*} x_1'(t) \amp = x_1(t)-x_2(t) \\ x_2'(t) \amp = x_1(t)+3x_2(t) \end{align*}

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