Elementary Differential Equations and Boundary Value Problems

Overview: This lesson introduces the process of "Mathematical Modeling" by translating physical phenomena, such as falling objects and population dynamics, into differential equations. Students will learn to use "Direction Fields" as a powerful tool for the qualitative analysis of these models.

Learning Outcomes:

• Formulate differential equations based on physical laws, specifically Newton’s Second Law for objects falling in the atmosphere.
• Construct and Interpret direction fields to visualize the behavior of solutions for first-order differential equations.
• Identify and Analyze equilibrium solutions and terminal velocity to determine the qualitative behavior of a system.

Overview: This lesson covers the fundamental theory, solution techniques, and practical applications of first-order differential equations. Master analytical methods like integrating factors and separation of variables, alongside numerical approximations through Euler’s method.

Learning Outcomes:

• Classify differential equations by order and determine linearity vs. nonlinearity.
• Solve first-order equations using integrating factors, separation of variables, and methods for exact or Bernoulli equations.
• Apply first-order ODEs to model physical phenomena such as mixing problems, radiocarbon dating, and cooling laws.

Overview: This lesson covers the theory and application of second-order linear differential equations, focusing on both homogeneous and nonhomogeneous forms. Solutions for equations with constant coefficients across various root types are explored and applied to mechanical and electrical systems.

Learning Outcomes:

• Solve second-order linear homogeneous equations with constant coefficients and verify the fundamental set of solutions using the Wronskian.
• Apply the Method of Undetermined Coefficients and Variation of Parameters to find particular solutions for nonhomogeneous equations.
• Model and analyze physical systems (vibrations and circuits) to identify phenomena such as resonance, beats, and transient/steady-state behaviors.

Overview: This lesson extends the theory of linear differential equations from the second order to the nth order. It establishes foundational existence and uniqueness theorems and provide systematic methods for solving higher-order equations with constant or variable coefficients.

Learning Outcomes:

• Determine the existence and uniqueness intervals for solutions to nth order linear initial value problems.
• Verify linear independence of functions using the Wronskian determinant and find fundamental sets of solutions.
• Construct general solutions for constant-coefficient homogeneous equations by identifying real, repeated, and complex roots of the characteristic polynomial.

Overview: This lesson explores the use of power series to solve second-order linear differential equations when solutions cannot be expressed in terms of elementary functions. Students will distinguish between point types and utilize the Method of Frobenius.

Learning Outcomes:

• Identify and Classify Points: Distinguish between ordinary, regular singular, and irregular singular points of a differential equation.
• Derive Series Solutions: Apply power series expansions and the Method of Frobenius to find general solutions and determine their radius of convergence.
• Analyze Special Functions: Define and solve classic equations (Airy, Hermite, Legendre, Bessel) and recognize their polynomial or transcendental solutions.

Overview: This lesson explores the Laplace transform as a powerful integral transform used to convert linear differential equations with initial conditions into algebraic equations. It addresses complex forcing functions including piecewise continuous functions and impulsive inputs.

Learning Outcomes:

• Define the Laplace transform and determine its existence based on piecewise continuity and exponential order.
• Solve second-order linear initial value problems (IVPs) by transforming them into the s-domain and applying inverse transforms.
• Represent and transform discontinuous forcing functions using the Heaviside (unit step) function and translation theorems.

Overview: This lesson explores the theory and application of systems of first-order linear differential equations. Use linear algebra tools such as eigenvalues and matrix exponentials to find solutions for complex system scenarios.

Learning Outcomes:

• Transform any n-th order linear differential equation into a system of n first-order equations.
• Solve homogeneous linear systems with constant coefficients using eigenvalues and eigenvectors, including cases with repeated roots.
• Construct fundamental matrices and utilize the matrix exponential \exp(\mathbf{A}t) and diagonalization to solve systems.

Overview: This lesson covers fundamental numerical techniques for solving ordinary differential equations, ranging from one-step methods to multistep predictor-corrector methods. It emphasizes the balance between approximation accuracy and numerical stability.

Learning Outcomes:

• Implement and compare the Euler (Explicit) and Backward Euler (Implicit) methods.
• Quantify and distinguish between local truncation, global truncation, and round-off errors.
• Apply multistep methods (Adams-Bashforth/Moulton) and Predictor-Corrector loops to improve order of accuracy.

Overview: This lesson explores the qualitative analysis of nonlinear autonomous systems using phase plane techniques. Progress from classifying linear systems to analyzing complex behaviors like limit cycles and chaos in the Lorenz system.

Learning Outcomes:

• Classify critical points of linear and nonlinear systems based on eigenvalues and phase portraits.
• Linearize autonomous nonlinear systems using Jacobian matrices to determine local stability.
• Apply Liapunov’s Second Method and the Poincaré–Bendixson Theorem to prove stability or the existence of periodic solutions.

Overview: This lesson introduces techniques for solving linear partial differential equations governing heat conduction, wave propagation, and steady-state temperature. Utilize Fourier series and separation of variables to transform PDEs into ODEs.

Learning Outcomes:

• Solve two-point boundary value problems and identify the conditions for unique, infinite, or no solutions.
• Expand periodic functions into Fourier series using Euler-Fourier formulas and identify even and odd function properties.
• Apply the method of separation of variables to solve the Heat Equation, Wave Equation, and Laplace’s Equation.

Overview: This lesson explores the theoretical framework of Sturm-Liouville (S-L) problems. It covers properties of self-adjoint operators, orthogonality of eigenfunctions, and the application of Green's functions to nonhomogeneous problems.

Learning Outcomes:

• Define and identify regular and singular Sturm-Liouville boundary value problems.
• Utilize Lagrange's Identity to prove the self-adjointness of operators and the orthogonality of eigenfunctions.
• Solve nonhomogeneous boundary value problems using eigenfunction expansions and Green’s functions.