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- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_225_Differential_Equations/08%3A_Laplace_Transforms/8.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/05%3A_Matrix_Methods_for_Dynamical_Systems/5.02%3A_The_Laplace_TransformThe Laplace Transform of a matrix of functions is simply the matrix of Laplace transforms of the individual elements. \[ \begin{align*} \mathscr{L}(B \textbf{x}+\textbf{g}) &= \mathscr{L}(B \textbf{x}...The Laplace Transform of a matrix of functions is simply the matrix of Laplace transforms of the individual elements. \[ \begin{align*} \mathscr{L}(B \textbf{x}+\textbf{g}) &= \mathscr{L}(B \textbf{x})+\mathscr{L}(\textbf{g}) \\[4pt] &= B \mathscr{L}(\textbf{x})+\mathscr{L}(\textbf{g}) \end{align*}\] We note that \((sI-B)^{-1}\) well defined except at the roots of the quadratic, \(s^{2}-4s+3\) determinant of \((sI-B)\) and is often referred to as the characteristic polynomial of \(B\).
- https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Reed)/07%3A_Laplace_Transforms/7.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/09%3A_Laplace_Transforms/9.06%3A_Convolution\[\begin{aligned}{\cal L}|\sin t|&={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}|\sin t|\,dt \\[5pt] &={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}\sin t\,dt\\[5pt] &={1\over 1-e^{-s\pi}}\left(\int_0^\pi e^{-st}\s...\[\begin{aligned}{\cal L}|\sin t|&={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}|\sin t|\,dt \\[5pt] &={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}\sin t\,dt\\[5pt] &={1\over 1-e^{-s\pi}}\left(\int_0^\pi e^{-st}\sin t\,dt+\int_\pi^\infty e^{-st}(0)\,dt\right)\\[5pt] &={1\over 1-e^{-s\pi}}{\cal L}\left(\sin t-\sin t {\cal U}(t-\pi)\right)\\[5pt] &={1\over 1-e^{-s\pi}}{\cal L}\left(\sin t+\sin (t-\pi){\cal U}(t-\pi)\right)\\[5pt] &={1\over 1-e^{-s\pi}}\left({1\over s^2+1}+{e^{-\pi s}\over s^2+1}\right)\\[5pt]…
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/08%3A_The_Eigenvalue_Problem/8.02%3A_The_Resolvent\[\frac{1}{s-b} = \frac{\frac{1}{s}}{1-\frac{b}{s}} = \frac{1}{s}+\frac{b}{s^2}+ \cdots +\frac{b^{n-1}}{s^n}+\frac{b^n}{s^n} \frac{1}{s-b} \nonumber\] \[(sI-B)^{-1} = s^{-1} \left(I-\frac{B}{s}\right)...\[\frac{1}{s-b} = \frac{\frac{1}{s}}{1-\frac{b}{s}} = \frac{1}{s}+\frac{b}{s^2}+ \cdots +\frac{b^{n-1}}{s^n}+\frac{b^n}{s^n} \frac{1}{s-b} \nonumber\] \[(sI-B)^{-1} = s^{-1} \left(I-\frac{B}{s}\right)^{-1} \left(\frac{1}{s}+\frac{B}{s^2}+ \cdots +\frac{B^{n-1}}{s^n}+\frac{B^n}{s^n}(sI-B)^{-1}\right) \nonumber\] \[(s_{2}I-B)^{-1}-(s_{1}I-B)^{-1} = (s_{2}I-B)^{-1}(s_{1}I-B-s_{2}I+B)(s_{1}I-B)^{-1} \nonumber\]
- https://math.libretexts.org/Courses/Red_Rocks_Community_College/MAT_2561_Differential_Equations_with_Engineering_Applications/08%3A_Laplace_Transforms/8.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
- https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Kravets)/07%3A_Laplace_Transforms/7.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/06%3A_Laplace_Transforms/6.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/05%3A_The_Laplace_Transform/5.02%3A_Transforms_of_Derivatives_and_ODEsThe procedure for linear constant coefficient equations is as follows. We take an ordinary differential equation in the time variable t . We apply the Laplace transform to transform the equation into ...The procedure for linear constant coefficient equations is as follows. We take an ordinary differential equation in the time variable t . We apply the Laplace transform to transform the equation into an algebraic (non differential) equation in the frequency domain. We solve the equation for X(s) . Then taking the inverse transform, if possible, we find x(t). Unfortunately, not every function has a Laplace transform, not every equation can be solved in this manner.
