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[Analysis] Notes and cleanup
Notes for tangent space computation and green's formula. Also includes fixes for the pagination as result of additions
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@@ -1,6 +1,6 @@
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\newsectionNoPB
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\subsection{Linear Differential Equations}
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An ODE is considered linear if and only if the $y$s are only scaled and not part of powers.\\
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An ODE is considered \bi{linear} if and only if the $y$s are only scaled and not part of powers.\\
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\compactdef{Linear differential equation of order $k$} (order = highest derivative)
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$y^{(k)} + a_{k - 1}y^{(k - 1)} + \ldots + a_1 y' + a_0 y = b$, with $a_i$ and $b$ functions in $x$.
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If $b(x) = 0 \smallhspace \forall x$, \bi{homogeneous}, else \bi{inhomogeneous}\\
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@@ -29,5 +29,5 @@ The homogeneous equation will then be all the elements of the set summed up.\\
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\item \bi{Separation of variables} For equations of form $y' = a(y) \cdot b(x)$ (Note: Not linear),
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we transform into $\frac{y'}{a(y)} = b(x)$ and then integrate by substituting $y'(x) dx = dy$, changing the variable of integration.
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Solution: $A(y) = B(x) + c$, with $A = \int \frac{1}{a}$ and $B(x) = \int b(x)$.
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To get final solution, solve for the above equation for $y$.
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To get final solution, solve the above equation for $y$.
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\end{itemize}
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