Differential Equations are crucial in expressing the laws of nature, providing a foundational understanding for many scientific and engineering disciplines. This course covers:
Understanding these concepts is essential for addressing complex problems encountered in various scientific fields.
This module focuses on the geometrical perspective of first-order ordinary differential equations (ODEs) described by the equation y' = f(x,y). Students will learn how to analyze direction fields and integral curves. The understanding of these concepts is paramount as they illustrate the behavior of solutions to ODEs graphically. Key topics include:
This module introduces Euler's numerical method and its generalizations for solving first-order ODEs represented by y' = f(x,y). Students will become familiar with the basic principles of numerical methods and how they can be employed to approximate solutions to differential equations. Key concepts covered will include:
This module covers the techniques for solving first-order linear ordinary differential equations (ODEs) and distinguishing between steady-state and transient solutions. Students will explore various methods for finding solutions, including:
This module introduces first-order substitution methods, specifically Bernoulli and homogeneous ODEs. Students will learn how to apply substitution techniques to simplify and solve complex equations effectively. Key learning points include:
This module emphasizes the qualitative methods and applications of first-order autonomous ODEs. Students will investigate the behavior of solutions without necessarily finding explicit solutions. Important topics include:
This module introduces complex numbers and complex exponentials as essential tools in solving differential equations. Students will learn how these concepts relate to ODEs and their applications, including:
This module discusses first-order linear ODEs with constant coefficients, examining methods for solving these equations and their applications. Students will explore:
This module covers various applications of ODEs in modeling real-world phenomena such as temperature changes, mixing processes, and circuit behaviors. Students will learn to set up and solve ODEs in practical contexts, focusing on:
This module teaches students how to solve second-order linear ODEs with constant coefficients. Emphasis will be placed on methods of solving these equations and their applications. Key topics include:
This module introduces the concept of complex characteristic roots and explores undamped and damped oscillations. Students will investigate the behavior of solutions under different conditions, covering:
This module focuses on second-order linear homogeneous ODEs, emphasizing the principle of superposition, uniqueness of solutions, and the use of Wronskians to analyze solutions. Key topics include:
This module examines inhomogeneous ODEs, focusing on methods for solving them and establishing stability criteria for constant-coefficient ODEs. Students will learn about:
This module covers inhomogeneous ODEs and introduces operator and solution formulas involving exponentials. Students will learn how to utilize these tools to solve complex equations effectively. Key topics include:
This module interprets the exceptional case of resonance in the context of differential equations. Students will learn how resonance affects the behavior of solutions and explore its implications in various applications, including:
This module introduces the fundamentals of Fourier series, including basic formulas for periodic functions defined over a period of 2Ï. Students will learn how to express functions as Fourier series and the significance of these representations, covering:
This module expands on Fourier series by discussing more general periods, even and odd functions, and periodic extensions. Students will learn how to apply these concepts to broaden their understanding of Fourier analysis, including:
This module focuses on finding particular solutions to differential equations using Fourier series, particularly emphasizing resonant terms. Students will learn how to apply Fourier series to solve ODEs effectively, covering:
This module introduces derivative formulas and the use of the Laplace transform to solve linear ODEs. Students will learn how to apply the Laplace transform as a powerful tool for solving differential equations, including:
This module covers the convolution formula, including proof, connections with the Laplace transform, and its applications in solving ODEs. Students will develop a deep understanding of convolution and its significance in systems analysis, focusing on:
This module explores the use of the Laplace transform to solve ODEs with discontinuous inputs. Students will learn how to handle discontinuities effectively in differential equations, covering:
This module delves into the concept of impulse inputs and the Dirac Delta function, which are crucial in understanding dynamic systems and their responses. The Dirac Delta function serves as a mathematical representation of an idealized impulse, enabling engineers and scientists to model systems subject to sudden forces. Key topics covered include:
By the end of this module, students will appreciate the role of impulse inputs in system analysis and response characterization.
This module focuses on first-order systems of ordinary differential equations (ODEs) and their solutions. Students will learn various methods for solving these systems, including:
By the end of this module, learners will be equipped to analyze and solve first-order ODEs with confidence.
In this module, students will explore homogeneous linear systems with constant coefficients. The focus will be on solving these systems using matrix eigenvalues, which are fundamental in analyzing system behavior. Key topics include:
This module provides essential techniques for approaching linear systems, particularly in engineering contexts.
This module continues the study of linear systems, focusing on repeated real eigenvalues and complex eigenvalues. Students will learn how to handle cases where eigenvalues are not distinct, and how complex eigenvalues influence system behavior. Topics include:
By the end of this module, students will have a comprehensive understanding of advanced eigenvalue concepts.
This module covers the sketching of solutions for 2x2 homogeneous linear systems with constant coefficients. Students will learn how to visualize and interpret the behavior of these systems. Key elements include:
By mastering these skills, students will enhance their ability to analyze linear systems effectively.
This module introduces matrix methods for solving inhomogeneous systems. Students will learn how to apply matrix techniques to find particular solutions, with a focus on real-world applications. Important topics include:
The knowledge gained will empower students to tackle more complex differential equations.
This module delves into the concept of matrix exponentials and their applications in solving systems of differential equations. Key areas of focus include:
Students will leave this module with practical tools for solving complex systems efficiently.
This module discusses the decoupling of linear systems with constant coefficients. Students will learn methods to transform coupled systems into decoupled forms, facilitating easier analysis and solutions. Key topics include:
Upon completion, students will have valuable skills for handling complex systems in engineering contexts.
This module introduces non-linear autonomous systems, focusing on finding critical points and sketching trajectories. Students will gain insight into the behavior of non-linear systems and their dynamics. Topics covered include:
By mastering these concepts, students will be able to analyze and predict the behavior of non-linear systems effectively.
This module focuses on limit cycles, addressing both existence and non-existence criteria. Students will learn the conditions under which limit cycles arise in non-linear systems. Key areas of study include:
Students will complete this module with a solid understanding of limit cycles and their implications in system behavior.
This module examines non-linear systems and their relationship with first-order ordinary differential equations (ODEs). Students will explore the methods for analyzing these systems and their solutions. Key topics include:
By the end of this module, students will be adept at tackling non-linear systems within various contexts.