Explore the core concepts of Ordinary and Partial Differential Equations (ODE/PDE), Fourier Series, and Hilbert Spaces with this concise lecture notes collection from the University of Siena. Designed as a supplementary text, it offers an informal yet informative introduction to essential topics in Mathematical Physics. Delve into higher-order ODEs, classical PDE methods like separation of variables, and the role of Fourier series. The book also provides foundational insights into Hilbert spaces and orthonormal expansions, crucial for a solid understanding of these advanced mathematical tools. Ideal for students seeking to complement their calculus and mathematical analysis background, this textbook is a valuable resource for a focused and efficient review. With clear explanations and relevant appendices, it bridges the gap to more comprehensive studies in these areas.