polish: cross-reference verification and final formatting

- Fix overfull hbox in ch05 method comparison table (118pt -> resolved)
- Fix overfull hbox in ch09 verification equation (147pt -> split with align)
- Fix overfull hbox in ch08 phase plane table (16pt -> p-column width)
- Fix overfull hbox in appA VOP formula (70pt -> inline textstyle)
- Fix overfull hbox in appB Fourier series table (70pt -> resizebox)
- Fix overfull hbox in appB common integrals table (tabular width)
- Fix overfull hbox in appB Laplace properties table (tabular width)
- Fix float too large in ch06 TikZ resonance figure (scale + clip)
- Fix text overfull in ch03 terminal velocity paragraph
- Fix ch09 summary table width with @{} column specifiers

appendices/appA_solution_summary.tex

@@ -89,11 +89,9 @@ 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 $W = y_1 y_2' - y_2 y_1'$. &
+$\textstyle y_p(x) = -y_1(x)\!\int \frac{y_2(x)\,g(x)}{a\,W(x)}\,\diff x + y_2(x)\!\int \frac{y_1(x)\,g(x)}{a\,W(x)}\,\diff x$
+
+where $\textstyle W = y_1 y_2' - y_2 y_1'$. &
 Any $g(x)$ for which the integrals can be evaluated; requires the fundamental set $\{y_1,y_2\}$ \\[8pt]
 \bottomrule
 \end{tabular}

appendices/appB_transform_tables.tex

@@ -21,7 +21,7 @@ provided the integral converges. The following table lists the most frequently e
 \centering
 \caption{Common Laplace Transforms}
 \label{tab:laplace_transforms}
-\begin{tabular}{l l l}
+\begin{tabular}{l l p{3cm}}
 \toprule
 \textbf{$f(t)$} & \textbf{$\mathcal{L}\{f(t)\} = F(s)$} & \textbf{Conditions} \\
 \midrule
@@ -56,7 +56,7 @@ The following algebraic and operational properties make the Laplace transform a
 \centering
 \caption{Laplace Transform Properties}
 \label{tab:laplace_properties}
-\begin{tabular}{l l}
+\begin{tabular}{@{}l p{12cm}@{}}
 \toprule
 \textbf{Property} & \textbf{Formula} \\
 \midrule
@@ -90,11 +90,13 @@ Fourier series decompose a periodic function $f(x)$ of period $2L$ into sine and
 \centering
 \caption{Fourier Series on $[-L, L]$}
 \label{tab:fourier_series}
+\renewcommand{\arraystretch}{1.3}
+\resizebox{\textwidth}{!}{%
 \begin{tabular}{l l}
 \toprule
 \textbf{Formula} & \textbf{Expression} \\
 \midrule
-Full series & $\displaystyle f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \Bigl[a_n \cos\!\Bigl(\frac{n\pi x}{L}\Bigr) + b_n \sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\Bigr]$ \\[12pt]
+Full series & $\displaystyle f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \Bigl[a_n \cos\!\Bigl(\frac{n\pi x}{L}\Bigr) + b_n \sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\Bigr]$ \\[10pt]
 $a_0$ & $\displaystyle a_0 = \frac{1}{L}\int_{-L}^{L} f(x)\,\diff x$ \\[8pt]
 $a_n$ & $\displaystyle a_n = \frac{1}{L}\int_{-L}^{L} f(x)\cos\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt]
 $b_n$ & $\displaystyle b_n = \frac{1}{L}\int_{-L}^{L} f(x)\sin\!\Bigl(\frac{n\pi x}{L}\Bigr)\,\diff x$ \\[8pt]
@@ -104,7 +106,8 @@ Complex coefficients & $\displaystyle c_n = \frac{1}{2L}\int_{-L}^{L} f(x)\,e^{-
 Complex series & $\displaystyle f(x) = \sum_{n=-\infty}^{\infty} c_n\,e^{i n\pi x/L}$ \\[8pt]
 Parseval's identity & $\displaystyle \frac{1}{L}\int_{-L}^{L} |f(x)|^{2}\,\diff x = \frac{a_0^{2}}{2} + \sum_{n=1}^{\infty} \bigl(a_n^{2} + b_n^{2}\bigr)$ \\[8pt]
 \bottomrule
-\end{tabular}
+\end{tabular}%
+}
 \end{table}
 
 \subsection{Common Integral Table}
@@ -116,7 +119,7 @@ The following integrals are used throughout the handbook, particularly in separa
 \centering
 \caption{Common Indefinite Integrals}
 \label{tab:common_integrals}
-\begin{tabular}{l l}
+\begin{tabular}{@{}l p{11.5cm}@{}}
 \toprule
 \textbf{Integrand} & \textbf{Result} \\
 \midrule

chapters/ch03_qualitative.tex

@@ -612,7 +612,7 @@ Since $f'(v^*) = -k/m < 0$, the terminal velocity is asymptotically stable: any
   \label{eq:falling_quadratic}
   m\,\frac{\mathrm{d}v}{\mathrm{d}t} = mg - kv^2.
 \end{equation}
-The terminal velocity is $v^* = \sqrt{mg/k}$. The qualitative behavior is the same---solutions converge to $v^*$ asymptotically---but the approach is different from the linear case.
+The terminal velocity is $v^* = \sqrt{mg/k}$. The qualitative behavior is the same: solutions converge to $v^*$ asymptotically. The approach, however, is different from the linear case.
 
 \subsection{Summary}
 \label{sec:ch03_summary}

chapters/ch05_second_order_nonhomogeneous.tex

@@ -723,17 +723,17 @@ Both undetermined coefficients (UC) and variation of parameters (VoP) find parti
 \centering
 \caption{Undetermined coefficients vs.\ variation of parameters}
 \label{tab:ch05_method_comparison}
-\begin{tabular}{l l l}
+\begin{tabular}{p{2.8cm} p{4.8cm} p{6.2cm}}
   \toprule
   \textbf{Aspect} & \textbf{Undetermined Coefficients} & \textbf{Variation of Parameters} \\
   \midrule
-  Applicable $g(x)$ & Polynomials, $e^{\alpha x}$, $\sin(\beta x)$, $\cos(\beta x)$, and their sums/products & \emph{Any} continuous $g(x)$ (in principle) \\
-  Requires $y_h$? & Only for overlap check (modification rule) & Yes --- need a full fundamental set \\
+  Applicable $g(x)$ & Polynomials, $e^{\alpha x}$, $\sin/\cos(\beta x)$, sums/products & \emph{Any} continuous $g(x)$ (in principle) \\
+  Requires $y_h$? & Only for overlap check & Yes --- need a full fundamental set \\
   Requires $W(x)$? & No & Yes \\
   Normalized form? & Not required & \textbf{Required} (coefficient of $y''$ must be~1) \\
   Computation & Algebra (solve linear system) & Integration (may be difficult or non-elementary) \\
   Speed (simple $g(x)$) & Fast --- algebraic, no integrals & Slower --- integrals needed \\
-  Speed (complex $g(x)$) & \textbf{Not applicable} --- method fails & Always applicable (if integrals exist) \\
+  Speed (complex $g(x)$) & \textbf{N/A} --- method fails & Always applicable (if integrals exist) \\
   \bottomrule
 \end{tabular}
 \end{table}

chapters/ch06_mechanical_applications.tex

@@ -492,7 +492,7 @@ The amplitude $C(\omega)$ as a function of the forcing frequency is called the \
 
 \begin{figure}[htbp]
 \centering
-\begin{tikzpicture}[scale=0.9]
+\begin{tikzpicture}[scale=0.8, clip]
   \draw[->] (0,0) -- (6.5,0);
   \draw[->] (0,-0.3) -- (0,3.5);
   \node[below] at (6.5,0) {\small $\omega$};
@@ -507,8 +507,10 @@ The amplitude $C(\omega)$ as a function of the forcing frequency is called the \
   \draw[thick, blue!70] plot[domain=0.1:6.2, samples=300, smooth]
     ({\x}, {2.0 / sqrt((1 - 0.0816*\x*\x)^2 + (0.15*\x)^2)});
 
-  % Undamped curve (narrower, higher peak)
-  \draw[thick, red!60, dashed] plot[domain=0.1:6.2, samples=300, smooth]
+  % Undamped curve (narrower, higher peak) -- split around singularity at omega~3.51
+  \draw[thick, red!60, dashed] plot[domain=0.1:3.35, samples=150, smooth]
+    ({\x}, {2.0 / abs(1 - 0.0816*\x*\x)});
+  \draw[thick, red!60, dashed] plot[domain=3.65:6.2, samples=150, smooth]
     ({\x}, {2.0 / abs(1 - 0.0816*\x*\x)});
 
   % Legend

chapters/ch08_systems.tex

@@ -518,7 +518,7 @@ For $2 \times 2$ systems $\mathbf{x}' = A\mathbf{x}$, the qualitative behavior o
   \textbf{Phase plane classification for $2 \times 2$ systems $\mathbf{x}' = A\mathbf{x}$.}
 
   \begin{center}
-  \begin{tabular}{l l}
+  \begin{tabular}{l p{10cm}}
     \toprule
     \textbf{Eigenvalues} & \textbf{Classification} \\
     \midrule

chapters/ch09_series_solutions.tex

@@ -317,9 +317,11 @@ The repeated-root case is analogous to the repeated-root case for constant-coeff
 
   \textit{Verification.} Compute $y' = 3c_1 x^2 - c_2 x^{-2}$ and $y'' = 6c_1 x + 2c_2 x^{-3}$. Substitute into the ODE:
   \[
-    x^2(6c_1 x + 2c_2 x^{-3}) - x(3c_1 x^2 - c_2 x^{-2}) - 3(c_1 x^3 + c_2 x^{-1})
-    = 6c_1 x^3 + 2c_2 x^{-1} - 3c_1 x^3 + c_2 x^{-1} - 3c_1 x^3 - 3c_2 x^{-1}
-    = (6 - 3 - 3)c_1 x^3 + (2 + 1 - 3)c_2 x^{-1} = 0. \quad \checkmark
+    \begin{split}
+    & x^2(6c_1 x + 2c_2 x^{-3}) - x(3c_1 x^2 - c_2 x^{-2}) - 3(c_1 x^3 + c_2 x^{-1}) \\
+    & \quad = 6c_1 x^3 + 2c_2 x^{-1} - 3c_1 x^3 + c_2 x^{-1} - 3c_1 x^3 - 3c_2 x^{-1} \\
+    & \quad = (6 - 3 - 3)c_1 x^3 + (2 + 1 - 3)c_2 x^{-1} = 0. \quad \checkmark
+    \end{split}
   \]
 \end{workedexample}
 
@@ -578,16 +580,16 @@ Cases~2 and~3 involve the logarithmic term $\ln(x)$, analogous to the repeated-r
 \centering
 \caption{Series solution methods: when to use and solution forms}
 \label{tab:ch09_summary}
-\begin{tabular}{l l p{4.5cm}}
+\begin{tabular}{@{}l l p{4.2cm}@{}}
 \toprule
 \textbf{Method} & \textbf{When to use} & \textbf{Solution form} \\
 \midrule
 Power series & $p(x)$ and $q(x)$ analytic at $x_0$ &
 $y(x) = \sum_{n=0}^{\infty} a_n (x-x_0)^n$ \\[12pt]
 Euler--Cauchy & $x^2 y'' + \alpha x y' + \beta y = 0$ &
-$y = c_1 x^{r_1} + c_2 x^{r_2}$ (and variants, see \cref{sec:ch09_euler_cauchy}) \\[12pt]
+$y = c_1 x^{r_1} + c_2 x^{r_2}$ (see \cref{sec:ch09_euler_cauchy}) \\[12pt]
 Frobenius & $x_0$ is a regular singular point; $xp(x)$ and $x^2q(x)$ analytic at $x_0$ &
-$y(x) = x^r \sum_{n=0}^{\infty} a_n x^n$ (with $r$ from the indicial equation) \\
+$y(x) = x^r \sum_{n=0}^{\infty} a_n x^n$ \\
 \bottomrule
 \end{tabular}
 \end{table}