\contentsline {part}{I\hspace {1em}Fundamentals}{1}{part.1}%
\contentsline {chapter}{\numberline {1}Introduction to ODES}{3}{chapter.1}%
\contentsline {section}{\numberline {1.1}Types of differential equations}{5}{section.1.1}%
\contentsline {subsection}{\numberline {1.1.1}Order and degree}{5}{subsection.1.1.1}%
\contentsline {subsection}{\numberline {1.1.2}Forms of equations}{5}{subsection.1.1.2}%
\contentsline {section}{\numberline {1.2}First order separable ODE}{6}{section.1.2}%
\contentsline {section}{\numberline {1.3}Integrating factor}{6}{section.1.3}%
\contentsline {section}{\numberline {1.4}Second order linear, homogeneous, constant coefficient ODE}{7}{section.1.4}%
\contentsline {section}{\numberline {1.5}Exact equations}{9}{section.1.5}%
\contentsline {subsection}{\numberline {1.5.1}Solving by conservative functions}{9}{subsection.1.5.1}%
\contentsline {subsection}{\numberline {1.5.2}Solving by homogeneity}{10}{subsection.1.5.2}%
\contentsline {section}{\numberline {1.6}Bernoulli equations}{10}{section.1.6}%
\contentsline {section}{\numberline {1.7}Riccati equations}{11}{section.1.7}%
\contentsline {section}{\numberline {1.8}Basic tools for solving ODEs}{11}{section.1.8}%
\contentsline {section}{\numberline {1.9}Method of undetermined coefficients}{11}{section.1.9}%
\contentsline {section}{\numberline {1.10}Differential operators}{12}{section.1.10}%
\contentsline {section}{\numberline {1.11}Annihilator method}{12}{section.1.11}%
\contentsline {section}{\numberline {1.12}Picard iteration}{12}{section.1.12}%
\contentsline {chapter}{\numberline {2}Linear ODEs}{13}{chapter.2}%
\contentsline {section}{\numberline {2.1}Wronskian}{13}{section.2.1}%
\contentsline {subsection}{\numberline {2.1.1}Properties of the Wronskian}{14}{subsection.2.1.1}%
\contentsline {subsection}{\numberline {2.1.2}Abel's formula}{14}{subsection.2.1.2}%
\contentsline {section}{\numberline {2.2}Variation of Parameters (VOP)}{15}{section.2.2}%
\contentsline {section}{\numberline {2.3}Series methods}{15}{section.2.3}%
\contentsline {chapter}{\numberline {3}Laplace transform}{17}{chapter.3}%
\contentsline {chapter}{\numberline {4}PDEs}{21}{chapter.4}%
\contentsline {section}{\numberline {4.1}Separation of variables}{21}{section.4.1}%
\contentsline {section}{\numberline {4.2}Fourier transform method}{22}{section.4.2}%
\contentsline {section}{\numberline {4.3}Qualitative theorems}{23}{section.4.3}%
\contentsline {subsection}{\numberline {4.3.1}Maximum principle}{23}{subsection.4.3.1}%
\contentsline {subsection}{\numberline {4.3.2}Harnack's theorem}{23}{subsection.4.3.2}%
\contentsline {part}{II\hspace {1em}Advanced}{25}{part.2}%
\contentsline {chapter}{\numberline {5}Existence and uniqueness of differential equations}{27}{chapter.5}%
\contentsline {section}{\numberline {5.1}Peano existence theorem}{27}{section.5.1}%
\contentsline {section}{\numberline {5.2}Carathéodory existence theorem}{28}{section.5.2}%
\contentsline {section}{\numberline {5.3}Picard–Lindelöf theorem}{28}{section.5.3}%
\contentsline {subsection}{\numberline {5.3.1}Picard iteration}{28}{subsection.5.3.1}%
\contentsline {section}{\numberline {5.4}Cauchy-Kovalevskaya theorem}{28}{section.5.4}%
\contentsline {section}{\numberline {5.5}Okamura's theorem}{29}{section.5.5}%
\contentsline {chapter}{\numberline {6}Sturm-Liouville theory}{31}{chapter.6}%
\contentsline {chapter}{\numberline {7}Fuchsian theory}{33}{chapter.7}%
\contentsline {chapter}{\numberline {8}Lie theory}{35}{chapter.8}%