Unit 01
Differential Equations, Models & Qualitative Structure
U01-L01
Rates of Change and Differential-Equation Models
U01-L02
Classification: Order, Linearity and Autonomy
U01-L03
Solutions, Families and Initial-Value Problems
U01-L04
Direction Fields and Isoclines
U01-L05
Existence, Uniqueness and Maximal Intervals
U01-L06
Autonomous Equations, Equilibria and Stability
U01-L07
Units, Scaling, Assumptions and Model Validation
Unit 02
Separable Equations, Growth, Decay & Nonlinear First-Order Models
U02-L01
Separation of Variables
U02-L02
Exponential Growth and Decay
U02-L03
Doubling Time, Half-Life and Parameter Identification
U02-L04
Logistic Growth and Carrying Capacity
U02-L05
Threshold, Harvesting and Finite-Time Blow-Up
U02-L06
Homogeneous First-Order Substitution
U02-L07
Orthogonal Trajectories and Family Geometry
Unit 03
First-Order Linear, Exact & Transformable Equations
U03-L01
First-Order Linear Equations and Integrating Factors
U03-L02
Exact Equations and Potential Functions
U03-L03
Integrating Factors for Nonexact Equations
U03-L04
Bernoulli Equations
U03-L05
Substitutions for Shifted and Composite Slopes
U03-L06
Singular Solutions and Envelopes
U03-L07
First-Order Strategy and Interval Audit
Unit 04
Applied First-Order Models in Physics, Engineering & Life Science
U04-L01
Newton Cooling and Heating
U04-L02
Mixing Tanks and Conservation of Solute
U04-L03
Falling Bodies with Linear Drag
U04-L04
Population Harvesting and Threshold Management
U04-L05
Chemical Kinetics and Compartment Models
U04-L06
RC Circuits and Electrical Time Constants
U04-L07
Assumptions, Calibration and Sensitivity in Applied Models
Unit 05
Numerical Methods, Error & Computational Reliability
U05-L01
Euler’s Method from Tangent-Line Prediction
U05-L02
Improved Euler and Predictor–Corrector Reasoning
U05-L03
Classical Fourth-Order Runge–Kutta
U05-L04
Local Truncation, Global Error and Convergence
U05-L05
Absolute Stability and Stiff Equations
U05-L06
Adaptive Step Size and Event Detection
U05-L07
Reproducible Numerical Experiments
Unit 06
Second-Order Linear Homogeneous Equations & Mechanical Response
U06-L01
Second-Order Linear Structure and Characteristic Equations
U06-L02
Distinct Real Roots and Exponential Modes
U06-L03
Repeated Roots and Generalized Modes
U06-L04
Complex Roots and Oscillatory Motion
U06-L05
Initial Conditions, Amplitude and Phase
U06-L06
Wronskians and Fundamental Solution Sets
U06-L07
Reduction of Order and Operator Verification
Unit 07
Forced Equations, Oscillation, Resonance & Circuit Models
U07-L01
Undetermined Coefficients for Standard Inputs
U07-L02
Variation of Parameters
U07-L03
Mass–Spring Models and Natural Frequency
U07-L04
Damping Regimes and Transient Design
U07-L05
Sinusoidal Forcing and Resonance
U07-L06
RLC Circuits as Second-Order Systems
U07-L07
Frequency Response, Energy and Engineering Limits
Unit 08
Laplace Transforms, Discontinuous Inputs & Transfer Functions
U08-L01
Laplace Transform Definition and Basic Properties
U08-L02
Inverse Transforms and Partial Fractions
U08-L03
Transforms of Derivatives and Initial-Value Problems
U08-L04
Unit-Step Functions and Piecewise Inputs
U08-L05
Impulse Inputs and Jump Conditions
U08-L06
Convolution and Impulse Response
U08-L07
Transfer Functions, Poles and Stability
Unit 09
Systems of Differential Equations & Matrix Methods
U09-L01
From Higher-Order Equations to First-Order Systems
U09-L02
Linear Systems and Eigenmode Solutions
U09-L03
Distinct Real Eigenvalues and Phase Portraits
U09-L04
Complex Eigenvalues, Spirals and Centers
U09-L05
Repeated Eigenvalues and Defective Systems
U09-L06
Fundamental Matrices and Matrix Exponentials
U09-L07
Forced Linear Systems and Variation of Constants
Unit 10
Nonlinear Systems, Phase Planes & Stability
U10-L01
Phase Planes, Nullclines and Direction Fields
U10-L02
Jacobian Linearization Near Equilibria
U10-L03
Competing Species and Invariant Regions
U10-L04
Predator–Prey Models and Conserved Orbits
U10-L05
Nonlinear Oscillators and Energy Landscapes
U10-L06
Lyapunov Functions and Stability Certification
U10-L07
Bifurcations, Limit Cycles and Chaos Preview
Unit 11
Power-Series Solutions, Frobenius Methods & Special Functions
U11-L01
Power-Series Method at an Ordinary Point
U11-L02
Recurrences for Second-Order Equations
U11-L03
Variable Coefficients and Radius of Convergence
U11-L04
Frobenius Method and Indicial Equations
U11-L05
Euler–Cauchy Equations
U11-L06
Bessel and Legendre Foundations
U11-L07
Series Truncation, Numerical Comparison and Verification
Unit 12
Boundary-Value Problems, Fourier Methods & PDE Bridge
U12-L01
Boundary-Value Problems versus Initial-Value Problems
U12-L02
Eigenvalue Boundary Problems and Discrete Modes
U12-L03
Sturm–Liouville Structure and Orthogonality
U12-L04
Fourier Series and Periodic Representation
U12-L05
Heat Equation by Separation of Variables
U12-L06
Wave Equation and Modal Oscillation
U12-L07
Laplace Equation and Numerical PDE Bridge
Unit 13
Engineering Modelling, Parameter Estimation, Control & Capstone
U13-L01
Parameter Estimation from Dynamic Data
U13-L02
Nondimensionalization and Similarity
U13-L03
Sensitivity, Identifiability and Uncertainty
U13-L04
Solver Selection and Reproducible Simulation
U13-L05
Feedback, Stability and Control Interpretation
U13-L06
Capstone Question, Model and Evidence Plan
U13-L07
Capstone Computation, Verification and Technical Report
U13-L08
Oral Defense and Advanced Differential-Equations Bridge