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