Differential equations – Systems of linear differential equations

During the course Linear Algebra we have already seen systems of linear differential equations. We use the notation

\[\mathbf{x}(t)=A(t)\mathbf{x}(t)+\mathbf{g}(t)\quad\text{with}\quad\mathbf{x}(t)=\begin{pmatrix}x_1(t)\\\vdots\\x_n(t)\end{pmatrix}, \quad A=\begin{pmatrix}a_{11}(t)&\ldots&a_{1n}(t)\\\vdots&\ddots&\vdots\\a_{n1}(t)&\ldots&a_{nn}(t)\end{pmatrix} \quad\text{and}\quad\mathbf{g}(t)=\begin{pmatrix}g_1(t)\\\vdots\\g_n(t)\end{pmatrix}.\]

Such a system is called homogeneous if \(\mathbf{g}(t)\equiv\mathbf{0}\) and otherwise nonhomogeneous.

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Last modified on July 1, 2021