Tidied up groups notes, up to date with lectures

Author: Zentrik

.gitignore

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 *.synctex.gz
 *.auxlock
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+*.kra~
+*.synctex
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Differential Equations/Higher-Order Linear Differential Equations - Topic 4.pdf

@@ -12,6 +12,38 @@
 \begin{document}
 \maketitle
 
+\section{Syllabus}
+
+
+{\small
+  \noindent\textbf{Basic calculus}\\
+  Informal treatment of differentiation as a limit, the chain rule, Leibnitz's rule, Taylor series, informal treatment of $O$ and $o$ notation and l'H\^opital's rule; integration as an area, fundamental theorem of calculus, integration by substitution and parts.\hspace*{\fill}[3]
+
+  \vspace{5pt}
+  \noindent Informal treatment of partial derivatives, geometrical interpretation, statement (only) of symmetry of mixed partial derivatives, chain rule, implicit differentiation. Informal treatment of differentials, including exact differentials. Differentiation of an integral with respect to a parameter.\hspace*{\fill}[2]
+
+  \vspace{10pt}
+  \noindent\textbf{First-order linear differential equations}\\
+  Equations with constant coefficients: exponential growth, comparison with discrete equations, series solution; modelling examples including radioactive decay.
+
+  \vspace{5pt}
+  \noindent Equations with non-constant coefficients: solution by integrating factor.\hspace*{\fill}[2]
+
+  \vspace{10pt}
+  \noindent\textbf{Nonlinear first-order equations}\\
+  Separable equations. Exact equations. Sketching solution trajectories. Equilibrium solutions, stability by perturbation; examples, including logistic equation and chemical kinetics. Discrete equations: equilibrium solutions, stability; examples including the logistic map.\hspace*{\fill}[4]
+
+  \vspace{10pt}
+  \noindent\textbf{Higher-order linear differential equations}\\
+  Complementary function and particular integral, linear independence, Wronskian (for second-order equations), Abel's theorem. Equations with constant coefficients and examples including radioactive sequences, comparison in simple cases with difference equations, reduction of order, resonance, transients, damping. Homogeneous equations. Response to step and impulse function inputs; introduction to the notions of the Heaviside step-function and the Dirac delta-function. Series solutions including statement only of the need for the logarithmic solution.\hspace*{\fill}[8]
+
+  \vspace{10pt}
+  \noindent\textbf{Multivariate functions: applications}\\
+  Directional derivatives and the gradient vector. Statement of Taylor series for functions on $\mathbb{R}^n$. Local extrema of real functions, classification using the Hessian matrix. Coupled first order systems: equivalence to single higher order equations; solution by matrix methods. Non-degenerate phase portraits local to equilibrium points; stability.
+
+  \vspace{5pt}
+  \noindent Simple examples of first- and second-order partial differential equations, solution of the wave equation in the form $f(x + ct) + g(x - ct)$.\hspace*{\fill}[5]}
+
 \tableofcontents
 
 \include{Introduction}