This book is an extension of courses taught in a one-year course. It caters to both applied differential equations and pure mathematics enthusiasts, reflecting the authors' specific interests. Readers should have a foundational understanding of matrices and complex variable functions. The Lebesgue integral is discussed in Chapter. In chapters, a solid grasp of integration theory is required, but an alternative approach using the Riemann-Stieltjes integral is offered for those less familiar with the Lebesgue integral. The book does not explore the historical origins of the theories discussed and includes limited references. Some problems offer additional material not covered in the text, providing further learning opportunities. The book was created with support from a grant from the Office of Naval Research, for which the authors are deeply grateful.