Differential Equations are critical for expressing the laws of nature and understanding contemporary science and engineering. This course covers:
This module introduces the geometrical interpretation of first-order differential equations through direction fields and integral curves. Students will learn how to visualize solutions and understand the behavior of ODEs graphically.
In this module, we will delve into Euler's numerical method for solving first-order ODEs. This includes exploring its generalizations and understanding the implementation of numerical techniques to approximate solutions effectively.
This module focuses on solving first-order linear ODEs and distinguishing between steady-state and transient solutions. It provides insights into the long-term behavior of systems modeled by ODEs.
Here, students will learn about first-order substitution methods, including Bernoulli and homogeneous ODEs. This module emphasizes recognizing suitable substitution techniques to simplify solving complex ODEs.
This module covers first-order autonomous ODEs, examining qualitative methods and their applications. Students will analyze critical points and understand the stability of solutions in various contexts.
This module introduces complex numbers and complex exponentials. Students will learn how these concepts relate to the solutions of differential equations, particularly in oscillatory systems.
In this section, students will tackle first-order linear ODEs featuring constant coefficients. The focus will be on solving these equations using various methodologies and understanding their applications in real-world scenarios.
This module involves applications of differential equations to models such as temperature, mixing, RC circuits, decay, and growth. Students will apply their knowledge of ODEs to solve practical problems.
This module focuses on solving second-order linear ODEs with constant coefficients. Students will learn techniques to find general solutions and understand the implications of different types of roots.
Students will explore complex characteristic roots, focusing on undamped and damped oscillations. This module highlights the impact of damping on the behavior of oscillatory systems modeled by ODEs.
This module covers second-order linear homogeneous ODEs, emphasizing superposition, uniqueness, and the Wronskian determinant. Students will learn how to analyze the solutions' behavior and verify their uniqueness.
This section introduces inhomogeneous ODEs and explores stability criteria for constant-coefficient ODEs. Students will apply these concepts to analyze the stability of solutions effectively.
This module discusses inhomogeneous ODEs, focusing on operator and solution formulas involving exponentials. Students will learn techniques for solving these equations and their practical applications.
This module covers the interpretation of resonance in differential equations, particularly in the context of exceptional cases. Students will learn about the implications of resonance on solution behavior.
This module introduces Fourier series, providing basic formulas for periodic functions. Students will learn how to represent functions using Fourier series and analyze their applications in solving ODEs.
In this section, students will explore more general periods, the concepts of even and odd functions, and periodic extension. The emphasis will be on understanding the periodic behavior of functions.
This module focuses on finding particular solutions using Fourier series, including techniques for handling resonant terms. Students will develop skills for extracting specific solutions from series representations.
This module introduces derivative formulas and the use of Laplace transforms to solve linear ODEs. Students will understand how to apply these transforms to simplify and solve differential equations.
In this module, students will study the convolution formula, its proof, and its relationship with the Laplace transform. Additionally, they will explore practical applications of convolution in solving differential equations.
This module discusses using the Laplace transform to address ODEs with discontinuous inputs. Students will learn how to handle such inputs effectively and solve corresponding differential equations.
This module covers impulse inputs, focusing on the Dirac delta function, weight, and transfer functions. Students will analyze the effects of impulse inputs on system behavior described by differential equations.
This module introduces first-order systems of ODEs, emphasizing the solution by elimination and providing geometric interpretations. Students will visualize the solutions and develop an understanding of system dynamics.
In this section, students will analyze homogeneous linear systems with constant coefficients, focusing on solutions through matrix eigenvalues. The module will highlight the significance of eigenvalues in understanding dynamic systems.
This module continues the study of linear systems, focusing on repeated real and complex eigenvalues. Students will explore the implications of these eigenvalues on the system's behavior and solution stability.
This module focuses on sketching solutions of 2x2 homogeneous linear systems with constant coefficients. Students will learn graphical methods to represent the behavior and solutions of these systems effectively.
In this module, students will explore matrix methods for inhomogeneous systems, focusing on using matrices to solve differential equations. This module provides practical applications of linear algebra in ODEs.
This module covers matrix exponentials and their application to solving systems of differential equations. Students will understand the role of exponentials in dynamic systems and gain computational skills for applying these techniques.
This final module explores decoupling linear systems with constant coefficients. Students will learn methods to simplify complex systems into manageable components, enhancing their ability to analyze and solve ODEs.
This module addresses non-linear autonomous systems, focusing on finding critical points and sketching trajectories. Students will learn qualitative analysis techniques to understand the dynamics of non-linear systems.
This module delves into limit cycles, discussing existence and non-existence criteria. Students will explore the conditions under which limit cycles occur in non-linear systems, enhancing their understanding of system behavior.
This concluding module focuses on non-linear systems and first-order ODEs, reinforcing the connections between linear and non-linear dynamics. Students will apply their knowledge to analyze and solve complex systems effectively.