Differential Equations

The course begins with essential preliminaries, including techniques such as differentiating under the integral sign, the Newton-Raphson method, properties of even and odd functions, piecewise continuity, and related foundational tools.

Students then study first-order differential equations in depth. Core techniques include separation of variables, integrating factors, substitution methods, and reduction to linear form. Special classes of equations—homogeneous, exact, Bernoulli, Riccati, Clairaut, and d'Alembert equations—are explored alongside slope fields, phase lines, equilibrium classification, and qualitative analysis. Extensive modeling applications include mixture problems, exponential and logistic growth, inhibited growth and decay, falling bodies with linear and quadratic drag, gravitational models, and RL/RC circuits.

The course then develops the theory of linear differential equations. Students examine linear differential operators, superposition, reduction of order, linear independence, Abel's identity, and the structure of solution spaces. Techniques for solving second-order linear equations with constant coefficients are studied for distinct, repeated, and complex roots, as well as polynomial, exponential, and sinusoidal forcing. Cauchy-Euler equations and variation of parameters extend these methods to broader classes of equations. Applications include simple harmonic motion, damped and forced oscillators, resonance phenomena, and RCL circuit models.

Systems of differential equations are introduced through matrix methods and eigenvalue analysis. Students solve homogeneous and inhomogeneous linear systems using eigenvalues, eigenvectors, matrix exponentials, and fundamental matrices. Phase planes, phase portraits, stability classification, linearization near equilibria, and shifted systems are studied in detail. Modeling applications include the Lotka-Volterra predator-prey system and extensions with carrying capacity.

Laplace transforms provide an alternative framework for solving linear equations and systems. Topics include unit step functions, shifting theorems, smoothness properties, transforms of derivatives and integrals, inverse transforms, and the solution of initial value problems with time-delayed forcing.

Boundary value problems and Fourier series expand the focus to eigenvalue problems and orthogonal expansions. Students study eigenvalues and eigenfunctions, Fourier sine and cosine series, convergence properties, arbitrary period expansions, and the use of Fourier methods to solve initial and boundary value problems.

Finally, the course develops numerical and series-based approximation methods. Students construct Taylor and power series solutions, apply the method of Frobenius at regular singular points, and derive recurrence relations. Numerical techniques include Euler's method (standard and modified), higher-order methods such as RK4 and Adams-Bashforth-Milne schemes, implicit methods (Implicit Euler and trapezoidal), and stability and error analysis using big-O notation and order of accuracy.

• Solve first-order differential equations using separation of variables, integrating factors, substitution methods, and reduction to linear form.

• Analyze and solve special first-order equations, including homogeneous, exact, Bernoulli, Riccati, Clairaut, and d'Alembert equations.

• Interpret slope fields, construct phase lines, classify equilibrium solutions, and perform qualitative analysis of autonomous and nonautonomous systems.

• Model real-world phenomena using first-order equations, including mixture problems, exponential and logistic growth, falling bodies with drag, gravitational models, and electrical circuits.

• Apply the theory of linear differential operators, superposition, reduction of order, and Abel's identity to determine solution structure.

• Solve second-order linear equations with constant coefficients for distinct, repeated, and complex roots, including equations with polynomial, exponential, and sinusoidal forcing.

• Use variation of parameters to solve first-, second-, and higher-order linear equations.

• Model oscillatory systems, including simple harmonic, damped, and forced oscillators, resonance phenomena, and RCL circuits.

• Formulate and solve systems of linear differential equations using eigenvalues, eigenvectors, matrix exponentials, and fundamental matrices.

• Construct and interpret phase portraits, classify stability types, and linearize systems near equilibrium points.

• Apply Laplace transforms and inverse transforms to solve initial value problems and systems with discontinuous or time-delayed forcing.

• Solve boundary value problems and compute eigenvalues and eigenfunctions.

• Develop and apply Fourier sine and cosine series, analyze convergence, and use Fourier methods to solve differential equations.

• Construct Taylor and Frobenius series solutions about ordinary and regular singular points.

• Implement numerical methods including Euler, modified Euler, RK4, Adams-Bashforth-Milne methods, implicit Euler, and trapezoidal methods.

• Analyze truncation error, order of accuracy, and stability of numerical schemes using big-O notation.

This course equips students with a unified framework for analyzing continuous dynamical systems, bridging theory, modeling, and computation in preparation for advanced study in applied mathematics, physics, engineering, and related fields.