Differential Equations
Ordinary differential equations, Laplace transforms, systems, nonlinear dynamics and an introduction to PDEs, applied to vibrations, circuits, control and heat.
MATH-201 Year 2 3 credits 12 lessons · 6 h 44 min
Offered by HSM University · Engineering Core
What you'll learn
- Solve first-order separable and linear equations and model cooling, decay, draining, mixing, RC circuits and drag
- Solve second-order linear equations with constant coefficients, including forced response, resonance and beam deflection
- Analyse mass-spring-damper and RLC systems using natural frequency, damping ratio and dynamic amplification
- Use Laplace transforms, step and impulse functions and transfer functions to solve initial value problems
- Solve linear systems with eigenvalues, find vibration modes, classify equilibria and solve systems numerically
- Analyse autonomous first-order equations with phase lines, bifurcations and the existence and uniqueness theorem, and solve exact and Bernoulli equations
- Apply variation of parameters, Duhamel's integral, Cauchy-Euler equations and higher-order equations to forced response, thick cylinders and beams on foundations
- Solve boundary value and eigenvalue problems for fins, column buckling and vibrating cables, and use orthogonality of eigenfunctions
- Use convolution, block diagrams, feedback, frequency response, Bode plots and the Routh-Hurwitz test to analyse dynamic systems
- Analyse nonlinear systems by linearization, energy and Lyapunov methods, and identify limit cycles and bifurcations
- Solve the heat, wave and Laplace equations by separation of variables and Fourier series, and check them with finite differences
- Formulate, solve, verify and interpret differential equation models of engineering systems
Course content
12 lessons · 6 h 44 min total · final exam
- 1 First-order equations 39 min · quiz
- 2 First-order equations: equilibria, existence and special methods 34 min · quiz
- 3 Second-order linear equations 37 min · quiz
- 4 More methods for linear equations 35 min · quiz
- 5 Mechanical and electrical oscillators 33 min · quiz
- 6 Boundary value and eigenvalue problems 32 min · quiz
- 7 Laplace transforms 36 min · quiz
- 8 Laplace techniques: convolution, feedback and frequency response 36 min · quiz
- 9 Systems of ODEs 32 min · quiz
- 10 Nonlinear systems and the phase plane 30 min · quiz
- 11 Partial differential equations and Fourier series 32 min · quiz
- 12 Modelling case studies and course project 28 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
Ordinary differential equations, Laplace transforms, systems, nonlinear dynamics and an introduction to PDEs, applied to vibrations, circuits, control and heat.
Part of the Engineering stream, typically taken in year 2.