Ordinary Differential Equations and Laplace Transforms
A 15-part course from mathematical foundations and ODE solution methods through circuit modelling, Laplace analysis and transients.
- 01. Mathematical Language & Notation for ODEsVariables, derivatives, integrals, initial conditions, common EE inputs and ODE classification
- 02. Algebra, Exponentials, Logarithms & Complex NumbersAlgebraic manipulation and the exponential, logarithmic and complex-number tools used in ODE analysis
- 03. Differentiation & Integration TechniquesDerivative and integration techniques for time-domain signals, circuit variables and differential equations
- 04. First-Order Ordinary Differential EquationsFirst-order solution families, initial conditions and methods for one-state dynamical systems
- 05. Second-Order Linear ODEs & Natural ModesTwo-state dynamics, characteristic roots, natural modes and the role of two initial conditions
- 06. From Circuit Laws to First-Order ODE ModelsBuilding first-order circuit models from KCL, KVL, element laws and consistent sign conventions
- 07. Second-Order Circuit ODEs: RLC NetworksRLC dynamics, stored energy, natural frequency, damping and second-order circuit behaviour
- 08. General ODE Modelling of Lumped Electrical NetworksChoosing variables and independent KCL/KVL equations for larger lumped electrical networks
- 09. The Laplace Transform: From ODEs to AlgebraUnilateral Laplace transforms, initial conditions and systematic treatment of switching inputs
- 10. Inverse Laplace, Partial Fractions & Pole StructureStandard transform pairs, partial fractions and the time-domain meaning of poles
- 11. Solving Circuit ODEs in the s-DomainSolving transformed circuit equations with Laplace methods, initial conditions and inverse transforms
- 12. Ex I: First-Order RC & RL SwitchingWorked switching responses using state continuity, final values, time constants and slope checks
- 13. Ex II: Second-Order RLC TransientsWorked RLC transients with stored energy, modal coefficients and damping classification
- 14. Ex III: Ramps, Pulses & ExponentialsForced circuit responses to ramp, pulse and exponential inputs in the time and Laplace domains
- 15. Ex IV: Sinusoids, Resonance & ResponseSinusoidal forcing, resonance, steady-state response and the transient needed for the complete solution
Ordinary Differential Equations and Laplace Transforms