diff --git a/appendices/appA_solution_summary.tex b/appendices/appA_solution_summary.tex
index 282590a..f599aa9 100644
--- a/appendices/appA_solution_summary.tex
+++ b/appendices/appA_solution_summary.tex
@@ -3,26 +3,408 @@
 \section{Solution Method Summary}
 \label{app:solution_summary}
 
-This appendix provides quick-reference tables for solution methods across all chapter topics.
+This appendix provides comprehensive quick-reference tables for every solution method covered in the handbook. Use these tables to identify the correct method for a given equation type and to recall key formulas.
+
+\subsection{First-Order ODE Methods}
+\label{sec:appA_first_order}
 
 \begin{table}[htbp]
 \centering
-\caption{Solution Methods by Equation Type}
-\label{tab:solution_methods}
-\begin{tabular}{l l l}
+\caption{First-Order Solution Methods (see \cref{ch:first_order})}
+\label{tab:appA_first_order_methods}
+\begin{tabular}{p{2.5cm} p{4.5cm} p{4.5cm} p{3.5cm}}
 \toprule
-\textbf{Equation Type} & \textbf{Method} & \textbf{Key Formula} \\
+\textbf{Method} & \textbf{Equation Form} & \textbf{Solution / Procedure} & \textbf{When to Use} \\
 \midrule
-Separable ODE & Separation of variables & TBD \\
-Linear first-order & Integrating factor & TBD \\
-Exact equations & Potential function & TBD \\
-Bernoulli & Substitution & TBD \\
-Second-order homogeneous & Characteristic equation & TBD \\
-Second-order nonhomogeneous & Undetermined coefficients & TBD \\
-Second-order nonhomogeneous & Variation of parameters & TBD \\
-IVP with Laplace & Laplace transform & TBD \\
-Systems of ODEs & Eigenvalue method & TBD \\
-Series solutions & Power series & TBD \\
+Separable &
+$\displaystyle \frac{\diff y}{\diff x} = g(x)\,h(y)$ &
+$\displaystyle \int \frac{\diff y}{h(y)} = \int g(x)\,\diff x + C$ &
+RHS factors as $g(x)\cdot h(y)$ \\[8pt]
+Linear &
+$\displaystyle y' + p(x)y = g(x)$ &
+$y = \dfrac{\displaystyle \int \mu(x)g(x)\,\diff x + C}{\mu(x)}$, \qquad $\mu = \exp\!\bigl(\int p(x)\,\diff x\bigr)$ &
+Can be written in standard linear form \\[8pt]
+Exact &
+$M(x,y)\,\diff x + N(x,y)\,\diff y = 0$, \qquad $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$ &
+Find $\psi(x,y)$ such that $\dfrac{\partial \psi}{\partial x} = M$, $\dfrac{\partial \psi}{\partial y} = N$; solution is $\psi(x,y) = C$ &
+Exactness test $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$ passes \\[8pt]
+Bernoulli &
+$\displaystyle y' + p(x)y = g(x)\,y^n$ &
+Substitute $v = y^{1-n}$; equation becomes linear in $v$ &
+RHS is a power of $y$ times a function of $x$ \\[8pt]
+Homogeneous &
+$\displaystyle \frac{\diff y}{\diff x} = F\!\left(\frac{y}{x}\right)$ &
+Substitute $y = vx$; equation becomes separable in $v$ &
+RHS is a function of the ratio $y/x$ \\[8pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\subsection{Second-Order Linear Homogeneous (Constant Coefficients)}
+\label{sec:appA_second_order_hom}
+
+Consider the equation $a\,y'' + b\,y' + c\,y = 0$ with $a,b,c \in \R$, $a \neq 0$.
+\begin{table}[htbp]
+\centering
+\caption{Characteristic Equation Cases (see \cref{ch:second_order_homogeneous})}
+\label{tab:appA_homogeneous_cases}
+\begin{tabular}{p{3cm} p{3cm} p{6cm} p{3cm}}
+\toprule
+\textbf{Discriminant} & \textbf{Roots of } $ar^2+br+c=0$ & \textbf{General Solution} & \textbf{Phase / Behavior} \\
+\midrule
+$b^2 - 4ac > 0$ \small (distinct real) &
+$r_1 = \dfrac{-b+\sqrt{b^2-4ac}}{2a}$, \ $r_2 = \dfrac{-b-\sqrt{b^2-4ac}}{2a}$ &
+$y = c_1 e^{r_1 x} + c_2 e^{r_2 x}$ &
+Exponential growth/decay; overdamped \\[12pt]
+$b^2 - 4ac = 0$ \small (repeated) &
+$r = -\dfrac{b}{2a}$ &
+$y = c_1 e^{rx} + c_2 x\,e^{rx}$ &
+Critically damped \\[12pt]
+$b^2 - 4ac < 0$ \small (complex conjugate) &
+$r = \alpha \pm i\beta$, \qquad $\alpha = -\dfrac{b}{2a}$, \ $\beta = \dfrac{\sqrt{4ac-b^2}}{2a}$ &
+$y = e^{\alpha x}\bigl[c_1\cos(\beta x) + c_2\sin(\beta x)\bigr]$ &
+Damped oscillation; underdamped \\[8pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{keyresult}
+  \textbf{Superposition principle.} For $L[y]=0$ linear homogeneous, any linear combination of solutions is again a solution. A \emph{fundamental set} $\{y_1,y_2\}$ has $W(y_1,y_2)\neq 0$, and the general solution is $y = c_1 y_1 + c_2 y_2$.
+\end{keyresult}
+
+\subsection{Second-Order Linear Nonhomogeneous}
+\label{sec:appA_second_order_nonhom}
+
+Consider $a\,y'' + b\,y' + c\,y = g(x)$. The general solution is $y = y_h + y_p$, where $y_h$ is the homogeneous solution and $y_p$ is a particular solution.
+
+\begin{table}[htbp]
+\centering
+\caption{Nonhomogeneous Solution Methods (see \cref{ch:second_order_nonhomogeneous})}
+\label{tab:appA_nonhomogeneous_methods}
+\begin{tabular}{p{3cm} p{6cm} p{4cm}}
+\toprule
+\textbf{Method} & \textbf{Formula / Procedure} & \textbf{When to Use} \\
+\midrule
+Undetermined Coefficients &
+See \cref{tab:appA_undetermined_guess} for guess table. Plug $y_p$ into ODE, solve for unknown coefficients. &
+$g(x)$ is a polynomial, exponential, sine/cosine, or finite sums/products thereof \\[12pt]
+Variation of Parameters &
+\[
+  y_p(x) = -y_1(x)\int \frac{y_2(x)\,g(x)}{a\,W(x)}\,\diff x
+            + y_2(x)\int \frac{y_1(x)\,g(x)}{a\,W(x)}\,\diff x
+\]
+where $W = y_1 y_2' - y_2 y_1'$. &
+Any $g(x)$ for which the integrals can be evaluated; requires the fundamental set $\{y_1,y_2\}$ \\[8pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{table}[htbp]
+\centering
+\caption{Guess Table for Undetermined Coefficients (see \cref{ch:second_order_nonhomogeneous})}
+\label{tab:appA_undetermined_guess}
+\begin{tabular}{p{4.5cm} p{5.5cm}}
+\toprule
+\textbf{Form of } $g(x)$ & \textbf{Guess for } $y_p(x)$ \\
+\midrule
+$P_n(x)$ (polynomial of degree $n$) &
+$A_n x^n + A_{n-1}x^{n-1} + \cdots + A_1 x + A_0$ \\[8pt]
+$A\,e^{kx}$ &
+$C\,e^{kx}$ \\[6pt]
+$A\sin(\omega x)$ or $A\cos(\omega x)$ &
+$C_1\cos(\omega x) + C_2\sin(\omega x)$ \\[6pt]
+$A\,e^{kx}\sin(\omega x)$ or $A\,e^{kx}\cos(\omega x)$ &
+$e^{kx}\bigl[C_1\cos(\omega x) + C_2\sin(\omega x)\bigr]$ \\[8pt]
+Sums of the above &
+Sum of the corresponding guesses \\[8pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{keyresult}
+  \textbf{Modification rule.} If any term of the guess $y_p$ is already a solution of the homogeneous equation (i.e.\ duplicates a term in $y_h$), multiply the \emph{entire} corresponding group by $x$. If $x$-multiplication still duplicates, multiply by $x^2$, and so on.
+\end{keyresult}
+
+\subsection{Laplace Transform Summary}
+\label{sec:appA_laplace}
+
+\begin{table}[htbp]
+\centering
+\caption{Common Laplace Transforms (see \cref{ch:laplace_transforms})}
+\label{tab:appA_laplace_transforms}
+\begin{tabular}{l l}
+\toprule
+\textbf{$f(t)$} & \textbf{$\mathcal{L}\{f(t)\} = F(s)$} \\
+\midrule
+$1$ & $\dfrac{1}{s}$ \\[8pt]
+$t^n$ \quad ($n \in \N$) & $\dfrac{n!}{s^{n+1}}$ \\[8pt]
+$e^{at}$ & $\dfrac{1}{s-a}$ \\[8pt]
+$\sin(\omega t)$ & $\dfrac{\omega}{s^2+\omega^2}$ \\[8pt]
+$\cos(\omega t)$ & $\dfrac{s}{s^2+\omega^2}$ \\[8pt]
+$\sinh(at)$ & $\dfrac{a}{s^2-a^2}$ \\[8pt]
+$\cosh(at)$ & $\dfrac{s}{s^2-a^2}$ \\[8pt]
+$e^{at}\sin(\omega t)$ & $\dfrac{\omega}{(s-a)^2+\omega^2}$ \\[8pt]
+$e^{at}\cos(\omega t)$ & $\dfrac{s-a}{(s-a)^2+\omega^2}$ \\[8pt]
+$t\,\sin(\omega t)$ & $\dfrac{2\omega s}{(s^2+\omega^2)^2}$ \\[8pt]
+$t\,\cos(\omega t)$ & $\dfrac{s^2-\omega^2}{(s^2+\omega^2)^2}$ \\[8pt]
+$u(t-a)$ \small (unit step) & $\dfrac{e^{-as}}{s}$ \\[8pt]
+$\delta(t-a)$ \small (Dirac delta) & $e^{-as}$ \\[8pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{table}[htbp]
+\centering
+\caption{Laplace Transform Properties (see \cref{ch:laplace_transforms})}
+\label{tab:appA_laplace_properties}
+\begin{tabular}{p{3.2cm} p{7.5cm}}
+\toprule
+\textbf{Property} & \textbf{Formula} \\
+\midrule
+Linearity &
+$\mathcal{L}\{a\,f(t) + b\,g(t)\} = a\,F(s) + b\,G(s)$ \\[10pt]
+First Shift (exponential) &
+$\mathcal{L}\{e^{at}f(t)\} = F(s-a)$ \\[10pt]
+Second Shift (time delay) &
+$\mathcal{L}\{u(t-a)\,f(t-a)\} = e^{-as}F(s)$ \\[10pt]
+Derivatives &
+$\mathcal{L}\{f'(t)\} = sF(s) - f(0)$ \\[4pt]
+& $\mathcal{L}\{f''(t)\} = s^2 F(s) - s\,f(0) - f'(0)$ \\[4pt]
+& $\mathcal{L}\{f^{(n)}(t)\} = s^n F(s) - s^{n-1}f(0) - \cdots - f^{(n-1)}(0)$ \\[10pt]
+Multiplication by $t^n$ &
+$\mathcal{L}\{t^n f(t)\} = (-1)^n \dfrac{\mathrm{d}^n}{\mathrm{d}s^n}F(s)$ \\[10pt]
+Convolution &
+$\mathcal{L}\{f * g\} = F(s)\,G(s)$, \quad $(f*g)(t) = \displaystyle\int_0^t f(\tau)\,g(t-\tau)\,\diff\tau$ \\[10pt]
+Integral &
+$\displaystyle \mathcal{L}\!\left\{\int_0^t f(\tau)\,\diff\tau\right\} = \frac{F(s)}{s}$ \\[10pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{keyresult}
+  \textbf{Solving IVPs with Laplace.} (1) Take the Laplace transform of the ODE (using derivative formulas to incorporate initial conditions). (2) Solve the resulting algebraic equation for $Y(s)$. (3) Find $y(t) = \mathcal{L}^{-1}\{Y(s)\}$ via partial fractions and the transform table.
+\end{keyresult}
+
+\subsection{Systems of Linear ODEs}
+\label{sec:appA_systems}
+
+For $\mathbf{x}' = A\,\mathbf{x}$ with $A$ a constant $n \times n$ matrix (see \cref{ch:systems}):
+
+\begin{table}[htbp]
+\centering
+\caption{Eigenvalue Cases for 2$\times$2 Systems (see \cref{ch:systems})}
+\label{tab:appA_systems_cases}
+\begin{tabular}{p{2.8cm} p{3.5cm} p{6.5cm}}
+\toprule
+\textbf{Eigenvalue Case} & \textbf{Eigenvalues / Vectors} & \textbf{General Solution} \\
+\midrule
+Real distinct &
+$\lambda_1 \neq \lambda_2 \in \R$; eigenvectors $\mathbf{v}_1, \mathbf{v}_2$ &
+$\mathbf{x}(t) = c_1 e^{\lambda_1 t}\,\mathbf{v}_1 + c_2 e^{\lambda_2 t}\,\mathbf{v}_2$ \\[12pt]
+Complex conjugate &
+$\lambda = \alpha \pm i\beta$; eigenvector $\mathbf{a} \pm i\mathbf{b}$ &
+$\mathbf{x}(t) = c_1\bigl[e^{\alpha t}\cos(\beta t)\,\mathbf{a} - e^{\alpha t}\sin(\beta t)\,\mathbf{b}\bigr]$
+$+ c_2\bigl[e^{\alpha t}\sin(\beta t)\,\mathbf{a} + e^{\alpha t}\cos(\beta t)\,\mathbf{b}\bigr]$ \\[12pt]
+Repeated (diagonalizable) &
+$\lambda_1=\lambda_2=\lambda$; two independent eigenvectors $\mathbf{v}_1,\mathbf{v}_2$ &
+$\mathbf{x}(t) = c_1 e^{\lambda t}\,\mathbf{v}_1 + c_2 e^{\lambda t}\,\mathbf{v}_2$ \\[10pt]
+Repeated (defective) &
+$\lambda$ (single eigenvector $\mathbf{v}_1$; generalized eigenvector $\mathbf{v}_2$) &
+$\mathbf{x}(t) = c_1 e^{\lambda t}\,\mathbf{v}_1 + c_2 e^{\lambda t}\bigl(t\,\mathbf{v}_1 + \mathbf{v}_2\bigr)$ \\[8pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{keyresult}
+  \textbf{Matrix exponential.} The solution to $\mathbf{x}' = A\mathbf{x},\; \mathbf{x}(0)=\mathbf{x}_0$ is $\mathbf{x}(t) = e^{At}\,\mathbf{x}_0$, where $e^{At} = \displaystyle\sum_{k=0}^{\infty} \frac{(At)^k}{k!}$. When $A$ is diagonalizable, $e^{At} = P\,e^{\Lambda t}\,P^{-1}$ with $\Lambda$ diagonal.
+\end{keyresult}
+
+\subsection{Series Solutions}
+\label{sec:appA_series}
+
+\begin{table}[htbp]
+\centering
+\caption{Series Solution Methods (see \cref{ch:series_solutions})}
+\label{tab:appA_series_methods}
+\begin{tabular}{p{2.5cm} p{3.8cm} p{5.5cm} p{3.5cm}}
+\toprule
+\textbf{Method} & \textbf{Equation Type} & \textbf{Solution Form} & \textbf{Key Details} \\
+\midrule
+Power series &
+$p(x)$ and $q(x)$ analytic at ordinary point $x_0$ in \ $y''+p(x)y'+q(x)y=0$ &
+$y = \sum_{n=0}^{\infty} a_n (x-x_0)^n$ &
+Substitute into ODE, equate coefficients to find recurrence for $a_n$ \\[12pt]
+Euler--Cauchy &
+$x^2 y'' + \alpha x y' + \beta y = 0$, \ $x > 0$ &
+Try $y = x^r$; indicial equation $r(r-1)+\alpha r+\beta = 0$ &
+Distinct roots $r_1\neq r_2$: $y=c_1 x^{r_1}+c_2 x^{r_2}$. Repeated: $y=x^r(c_1+c_2\ln x)$ \\[12pt]
+Frobenius &
+Regular singular point at $x_0$ in $y''+p(x)y'+q(x)y=0$ &
+$y = x^r\sum_{n=0}^{\infty} a_n x^n$, \quad $a_0\neq 0$ &
+Indicial equation from lowest power. Cases depend on $r_1-r_2$: non-integer, zero, or positive integer \\[8pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\subsection{Nonlinear Systems}
+\label{sec:appA_nonlinear}
+
+\begin{table}[htbp]
+\centering
+\caption{Linearized Classification via Jacobian (see \cref{ch:nonlinear_systems})}
+\label{tab:appA_nonlinear_classification}
+\begin{tabular}{p{3.2cm} p{3cm} p{3cm} p{3.8cm}}
+\toprule
+\textbf{Type} & \textbf{Trace } $T=\mathrm{tr}(J)$ & \textbf{Determinant } $D=\det(J)$ & \textbf{Eigenvalue Pattern} \\
+\midrule
+Stable node & $T<0$ & $D>0$, $T^2-4D>0$ & $\lambda_1,\lambda_2 < 0$, real \\[6pt]
+Unstable node & $T>0$ & $D>0$, $T^2-4D>0$ & $\lambda_1,\lambda_2 > 0$, real \\[6pt]
+Saddle point & $-$ & $D<0$ & $\lambda_1 < 0 < \lambda_2$, real \\[6pt]
+Stable spiral & $T<0$ & $D>0$, $T^2-4D<0$ & $\alpha \pm i\beta$, $\alpha < 0$ \\[6pt]
+Unstable spiral & $T>0$ & $D>0$, $T^2-4D<0$ & $\alpha \pm i\beta$, $\alpha > 0$ \\[6pt]
+Center & $T=0$ & $D>0$ & $\pm i\beta$ (pure imaginary) \\[6pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{keyresult}
+  \textbf{Hartman--Grobman theorem.} Near a hyperbolic equilibrium point ($\Re(\lambda_i)\neq 0$), the nonlinear system is topologically conjugate to its linearization. The phase portrait of the linearized system accurately describes the local behavior.
+\end{keyresult}
+
+\subsection{PDE Solution Methods}
+\label{sec:appA_pde}
+
+\begin{table}[htbp]
+\centering
+\caption{PDE Methods and Solution Forms (see \cref{ch:heat_equation}, \cref{ch:wave_and_laplace})}
+\label{tab:appA_pde_methods}
+\begin{tabular}{p{3.2cm} p{2.5cm} p{2.5cm} p{6cm}}
+\toprule
+\textbf{Equation} & \textbf{Domain} & \textbf{Method} & \textbf{Solution Form} \\
+\midrule
+Heat equation \ $\displaystyle u_t = \alpha\,u_{xx}$ & $0<x<L$ &
+Separation of variables + Fourier series &
+$\displaystyle u(x,t) = \sum_{n=1}^{\infty} b_n \sin\!\Bigl(\frac{n\pi x}{L}\Bigr)
+  \exp\!\Bigl(-\alpha\,\frac{n^2\pi^2}{L^2}\,t\Bigr)$ \\[12pt]
+Wave equation \ $\displaystyle u_{tt} = c^2\,u_{xx}$ & $x \in \R$ (infinite) &
+d'Alembert's formula &
+$\displaystyle u(x,t) = \tfrac{1}{2}\bigl[f(x-ct)+f(x+ct)\bigr]
+  + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,\diff s$ \\[12pt]
+Wave equation \ $\displaystyle u_{tt} = c^2\,u_{xx}$ & $0<x<L$ (finite, Dirichlet) &
+Separation + Fourier sine series &
+$\displaystyle u(x,t) = \sum_{n=1}^{\infty} \sin\!\Bigl(\frac{n\pi x}{L}\Bigr)
+  \bigl[A_n\cos(\tfrac{n\pi c t}{L}) + B_n\sin(\tfrac{n\pi c t}{L})\bigr]$ \\[12pt]
+Laplace \ $\displaystyle u_{xx}+u_{yy}=0$ & Rectangle $0<x<a$, $0<y<b$ &
+Separation of variables &
+$\displaystyle u(x,y) = \sum_{n=1}^{\infty} A_n \sin\!\Bigl(\frac{n\pi x}{a}\Bigr)
+  \sinh\!\Bigl(\frac{n\pi y}{a}\Bigr)$ \\[10pt]
+Laplace \ $\displaystyle u_{rr}+\tfrac{1}{r}u_r+\tfrac{1}{r^2}u_{\theta\theta}=0$ & Disk $r<a$ &
+Polar separation &
+$\displaystyle u(r,\theta) = A_0 + \sum_{n=1}^{\infty} r^n\bigl[A_n\cos(n\theta)+B_n\sin(n\theta)\bigr]$ \\[8pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{keyresult}
+  \textbf{Separation of variables (general procedure).} For a PDE $F(x,t,u,u_x,u_t,\dots)=0$, assume $u(x,t)=X(x)T(t)$. Substitute and separate variables to obtain two ODEs, one for $X$ and one for $T$, linked by a separation constant. Solve the spatial ODE as a Sturm--Liouville eigenvalue problem (\cref{ch:boundary_value_problems}), then expand the initial/boundary data in the resulting eigenfunction basis.
+\end{keyresult}
+
+\subsection{Mechanical and Electrical Applications}
+\label{sec:appA_applications}
+
+\begin{table}[htbp]
+\centering
+\caption{Damping Cases for Spring-Mass Systems (see \cref{ch:mechanical_applications})}
+\label{tab:appA_damping}
+\begin{tabular}{p{3.2cm} p{2.8cm} p{4.5cm} p{4cm}}
+\toprule
+\textbf{Case} & \textbf{Condition} & \textbf{Solution $y(t)$} & \textbf{Behavior} \\
+\midrule
+Undamped & $c=0$ &
+$y(t) = C_1\cos(\omega_0 t) + C_2\sin(\omega_0 t)$, \ $\omega_0 = \sqrt{k/m}$ &
+Persistent oscillation \\[10pt]
+Underdamped & $0 < c < 2\sqrt{km}$ &
+$y(t) = e^{-ct/(2m)}\bigl[C_1\cos(\omega_d t) + C_2\sin(\omega_d t)\bigr]$, \ $\omega_d = \sqrt{\omega_0^2-(c/2m)^2}$ &
+Decaying oscillation \\[10pt]
+Critically damped & $c = 2\sqrt{km}$ &
+$y(t) = C_1 e^{-\omega_0 t} + C_2 t\,e^{-\omega_0 t}$ &
+Fastest decay without oscillation \\[10pt]
+Overdamped & $c > 2\sqrt{km}$ &
+$y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}$, \ $r_{1,2} = \dfrac{-c \pm \sqrt{c^2-4km}}{2m}$ &
+Slow non-oscillatory decay \\[8pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{table}[htbp]
+\centering
+\caption{Mechanical--Electrical Analogy (see \cref{ch:mechanical_applications})}
+\label{tab:appA_mech_elec}
+\begin{tabular}{l l l l}
+\toprule
+\textbf{Mechanical} & \textbf{Electrical (charge)} & \textbf{Mechanical} & \textbf{Electrical (flux)} \\
+\midrule
+Mass $m$ & Inductance $L$ & Mass $m$ & Capacitance $C$ \\
+Damping $c$ & Resistance $R$ & Damping $c$ & Conductance $G$ \\
+Spring constant $k$ & Inverse capacitance $1/C$ & Spring constant $k$ & Inverse inductance $1/L$ \\
+Force $F(t)$ & Voltage $E(t)$ & Force $F(t)$ & Current $I(t)$ \\
+Displacement $y(t)$ & Charge $q(t)$ & Velocity $v(t)$ & Flux $\phi(t)$ \\
+\bottomrule
+\end{tabular}
+\end{table}
+
+\subsection{Qualitative Analysis and Numerical Methods}
+\label{sec:appA_qualitative}
+
+\begin{table}[htbp]
+\centering
+\caption{Equilibrium Classification for $y' = f(y)$ (see \cref{ch:qualitative})}
+\label{tab:appA_equilibrium}
+\begin{tabular}{p{3.5cm} p{4cm} p{4.5cm}}
+\toprule
+\textbf{Condition at } $y^*$ (where $f(y^*)=0$) & \textbf{Stability} & \textbf{Phase Line Behavior} \\
+\midrule
+$f'(y^*) < 0$ & Asymptotically stable (sink) & Solutions near $y^*$ converge to $y^*$ \\[6pt]
+$f'(y^*) > 0$ & Unstable (source) & Solutions near $y^*$ diverge away from $y^*$ \\[6pt]
+$f'(y^*) = 0$ & Inconclusive (semi-stable or higher-order) & Requires higher-order analysis; may be a node or saddle \\[8pt]
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{table}[htbp]
+\centering
+\caption{Euler's Method for Numerical Approximation (see \cref{ch:qualitative})}
+\label{tab:appA_euler}
+\begin{tabular}{l l}
+\toprule
+\textbf{Algorithm} & \\
+\midrule
+Given $y' = f(t,y)$, $y(t_0) = y_0$, step size $h$: \\[6pt]
+\quad $t_{n+1} = t_n + h$ \\[4pt]
+\quad $y_{n+1} = y_n + h\,f(t_n, y_n)$ \\[10pt]
+Local truncation error: $\mathcal{O}(h^2)$ \\
+Global error: $\mathcal{O}(h)$ \\
+\bottomrule
+\end{tabular}
+\end{table}
+
+\subsection{Fourier Series Summary}
+\label{sec:appA_fourier}
+
+\begin{table}[htbp]
+\centering
+\caption{Fourier Series Coefficients (see \cref{ch:fourier_series})}
+\label{tab:appA_fourier}
+\begin{tabular}{l l}
+\toprule
+\textbf{Series} & \textbf{Coefficients} \\
+\midrule
+Full range $[-L,L]$: \ $f(x) \sim \dfrac{a_0}{2} + \displaystyle\sum_{n=1}^{\infty} \Bigl[a_n\cos(\tfrac{n\pi x}{L}) + b_n\sin(\tfrac{n\pi x}{L})\Bigr]$ &
+$\displaystyle a_0 = \frac{1}{L}\int_{-L}^{L} f(x)\,\diff x$ \\[8pt]
+& $\displaystyle a_n = \frac{1}{L}\int_{-L}^{L} f(x)\cos\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt]
+& $\displaystyle b_n = \frac{1}{L}\int_{-L}^{L} f(x)\sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[10pt]
+Even $f$ (cosine series): & $b_n = 0$, \ $\displaystyle a_n = \frac{2}{L}\int_{0}^{L} f(x)\cos\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[10pt]
+Odd $f$ (sine series): & $a_n = 0$, \ $\displaystyle b_n = \frac{2}{L}\int_{0}^{L} f(x)\sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt]
 \bottomrule
 \end{tabular}
 \end{table}