skeleton: 14 chapter stubs and 4 appendices
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Worker Agent
28
appendices/appA_solution_summary.tex
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appendices/appA_solution_summary.tex
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\appendix
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\section{Solution Method Summary}
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\label{app:solution_summary}
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This appendix provides quick-reference tables for solution methods across all chapter topics.
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\begin{table}[htbp]
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\centering
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\caption{Solution Methods by Equation Type}
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\label{tab:solution_methods}
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\begin{tabular}{l l l}
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\toprule
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\textbf{Equation Type} & \textbf{Method} & \textbf{Key Formula} \\
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\midrule
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Separable ODE & Separation of variables & TBD \\
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Linear first-order & Integrating factor & TBD \\
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Exact equations & Potential function & TBD \\
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Bernoulli & Substitution & TBD \\
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Second-order homogeneous & Characteristic equation & TBD \\
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Second-order nonhomogeneous & Undetermined coefficients & TBD \\
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Second-order nonhomogeneous & Variation of parameters & TBD \\
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IVP with Laplace & Laplace transform & TBD \\
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Systems of ODEs & Eigenvalue method & TBD \\
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Series solutions & Power series & TBD \\
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\bottomrule
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\end{tabular}
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\end{table}
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53
appendices/appB_transform_tables.tex
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appendices/appB_transform_tables.tex
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\section{Transform Tables}
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\label{app:transform_tables}
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This appendix provides Laplace and Fourier transform tables for quick reference.
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\subsection{Laplace Transform Table}
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\label{sec:appB_laplace_table}
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\begin{table}[htbp]
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\centering
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\caption{Common Laplace Transforms}
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\label{tab:laplace_transforms}
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\begin{tabular}{l l}
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\toprule
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\textbf{$f(t)$} & \textbf{$\mathcal{L}\{f(t)\} = F(s)$} \\
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\midrule
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$1$ & TBD \\
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$t^n$ & TBD \\
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$e^{at}$ & TBD \\
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$\sin(\omega t)$ & TBD \\
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$\cos(\omega t)$ & TBD \\
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$e^{at}\sin(\omega t)$ & TBD \\
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$e^{at}\cos(\omega t)$ & TBD \\
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$\sinh(at)$ & TBD \\
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$\cosh(at)$ & TBD \\
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$t^n e^{at}$ & TBD \\
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$t \sin(\omega t)$ & TBD \\
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$t \cos(\omega t)$ & TBD \\
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$u(t-a)$ (unit step) & TBD \\
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$\delta(t-a)$ (Dirac delta) & TBD \\
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\bottomrule
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\end{tabular}
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\end{table}
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\subsection{Fourier Transform Table}
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\label{sec:appB_fourier_table}
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\begin{table}[htbp]
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\centering
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\caption{Common Fourier Transforms}
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\label{tab:fourier_transforms}
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\begin{tabular}{l l}
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\toprule
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\textbf{$f(x)$} & \textbf{$\hat{f}(k)$} \\
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\midrule
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$e^{iax}$ & TBD \\
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$e^{-a|x|}$ & TBD \\
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$\mathrm{rect}(x)$ & TBD \\
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$\delta(x)$ & TBD \\
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Gaussian $e^{-ax^2}$ & TBD \\
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\bottomrule
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\end{tabular}
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\end{table}
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71
appendices/appC_integral_tables.tex
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appendices/appC_integral_tables.tex
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\section{Integral Tables}
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\label{app:integral_tables}
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This appendix provides commonly used integral formulas encountered throughout the handbook.
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\subsection{Basic Integrals}
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\label{sec:appC_basic}
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\begin{table}[htbp]
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\centering
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\caption{Basic Integral Formulas}
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\label{tab:integrals_basic}
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\begin{tabular}{l l}
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\toprule
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\textbf{Integrand} & \textbf{Result} \\
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\midrule
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$\displaystyle \int x^n \, dx$ & TBD \\
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$\displaystyle \int \frac{1}{x} \, dx$ & TBD \\
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$\displaystyle \int e^{ax} \, dx$ & TBD \\
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$\displaystyle \int a^x \, dx$ & TBD \\
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$\displaystyle \int \sin(ax) \, dx$ & TBD \\
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$\displaystyle \int \cos(ax) \, dx$ & TBD \\
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$\displaystyle \int \tan(ax) \, dx$ & TBD \\
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$\displaystyle \int \sec^2(ax) \, dx$ & TBD \\
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$\displaystyle \int \csc^2(ax) \, dx$ & TBD \\
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$\displaystyle \int \sec(ax)\tan(ax) \, dx$ & TBD \\
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$\displaystyle \int \csc(ax)\cot(ax) \, dx$ & TBD \\
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\bottomrule
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\end{tabular}
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\end{table}
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\subsection{Integrals Involving Exponential and Trigonometric Functions}
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\label{sec:appC_exptig}
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\begin{table}[htbp]
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\centering
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\caption{Exponential--Trigonometric Integrals}
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\label{tab:integrals_exptig}
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\begin{tabular}{l l}
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\toprule
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\textbf{Integrand} & \textbf{Result} \\
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\midrule
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$\displaystyle \int e^{ax}\sin(bx)\,dx$ & TBD \\
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$\displaystyle \int e^{ax}\cos(bx)\,dx$ & TBD \\
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$\displaystyle \int x e^{ax}\,dx$ & TBD \\
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$\displaystyle \int x \sin(ax)\,dx$ & TBD \\
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$\displaystyle \int x \cos(ax)\,dx$ & TBD \\
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$\displaystyle \int \ln(x)\,dx$ & TBD \\
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\bottomrule
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\end{tabular}
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\end{table}
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\subsection{Integrals Involving Inverse Trigonometric Functions}
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\label{sec:appC_inverse_trig}
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\begin{table}[htbp]
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\centering
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\caption{Inverse Trigonometric Integrals}
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\label{tab:integrals_inverse_trig}
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\begin{tabular}{l l}
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\toprule
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\textbf{Integrand} & \textbf{Result} \\
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\midrule
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$\displaystyle \int \frac{1}{\sqrt{a^2 - x^2}}\,dx$ & TBD \\
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$\displaystyle \int \frac{1}{a^2 + x^2}\,dx$ & TBD \\
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$\displaystyle \int \frac{1}{x\sqrt{x^2 - a^2}}\,dx$ & TBD \\
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$\displaystyle \int \sqrt{a^2 - x^2}\,dx$ & TBD \\
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$\displaystyle \int \frac{1}{\sqrt{x^2 + a^2}}\,dx$ & TBD \\
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\bottomrule
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\end{tabular}
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\end{table}
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37
appendices/appD_notation.tex
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appendices/appD_notation.tex
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\section{Notation Glossary}
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\label{app:notation}
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This appendix provides a glossary of notation used throughout the handbook.
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\nomenclature{$y$}{Dependent variable; unknown function}
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\nomenclature{$y(t)$}{Dependent variable as a function of time $t$}
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\nomenclature{$y(x)$}{Dependent variable as a function of spatial variable $x$}
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\nomenclature{$x$}{Independent variable (spatial)}
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\nomenclature{$t$}{Independent variable (time)}
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\nomenclature{$\lambda$}{Eigenvalue; parameter in characteristic equation}
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\nomenclature{$\omega$}{Angular frequency}
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\nomenclature{$\omega_0$}{Natural angular frequency}
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\nomenclature{$\alpha, \beta, \gamma$}{General constants; damping ratio}
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\nomenclature{$\delta$}{Dirac delta function}
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\nomenclature{$\mu$}{Separation constant; parameter}
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\nomenclature{$\theta$}{Angle; phase shift}
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\nomenclature{$\phi$}{Eigenfunction; angle}
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\nomenclature{$A, B, C$}{General constants of integration}
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\nomenclature{$a, b, c$}{Coefficients in differential equations}
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\nomenclature{$n, m, k$}{Integer indices}
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\nomenclature{$f(t)$}{Input/forcing function}
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\nomenclature{$F(s)$}{Laplace transform of $f(t)$}
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\nomenclature{$\mathcal{L}$}{Laplace transform operator}
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\nomenclature{$\mathcal{L}^{-1}$}{Inverse Laplace transform}
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\nomenclature{$\mathcal{F}$}{Fourier transform operator}
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\nomenclature{$u(t)$}{Unit step (Heaviside) function}
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\nomenclature{$y_h$}{Homogeneous solution}
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\nomenclature{$y_p$}{Particular solution}
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\nomenclature{$W(y_1, y_2)$}{Wronskian of $y_1$ and $y_2$}
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\nomenclature{$\mathbf{A}$}{Coefficient matrix in systems}
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\nomenclature{$\mathbf{x}(t)$}{State vector}
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\nomenclature{$I_n$}{Identity matrix of size $n$}
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\nomenclature{$e^{\mathbf{A}t}$}{Matrix exponential}
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\nomenclature{$\mathbf{u}, \mathbf{v}$}{Vectors in phase plane}
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\nomenclature{$u(t-a)$}{Shifted unit step function}
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\nomenclature{$*$}{Convolution operator}
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