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7.31. Ordinary Differential Equations (Mandatory)

7.31.1. Justification ↑ Back to top

The Ordinary Differential Equations course provides the mathematical tools to analyze systems that evolve with respect to a single independent variable. It focuses on the modeling of dynamic systems through first-order equations, higher-order equations, and systems of equations, using both analytical and numerical methods. It is fundamental for the understanding of mechanics, electrical circuits, and population models.

7.31.2. Generales Goals ↑ Back to top

1. Solve first-order and higher-order ordinary differential equations using analytical methods.
2. Apply the Laplace transform to solve initial value problems involving discontinuous functions.
3. Analyze the stability of linear and nonlinear systems of differential equations.
4. Implement numerical methods for approximating solutions in complex models.

7.31.3. Contribution to Outcomes ↑ Back to top

ABET-1) An ability to identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics. (Familiarity)

7.31.4. Content ↑ Back to top

7.31.4.1. First-Order Ordinary Differential Equations (ODEs) (20 hours) [Skills ABET-1] ↑ Back to top

Bibliography: (Boyce et al., 2017; Zill, 2018)

Topics

1. Modeling with first-order equations: growth, mixtures, and cooling

Learning Outcomes

1. Solve linear first-order ODEs using integrating factors [Usage]

7.31.4.2. Higher-Order Differential Equations and Systems (ODEs) (40 hours) [Skills ABET-1] ↑ Back to top

Bibliography: (Boyce et al., 2017; Zill, 2018)

Topics

1. Definition and properties of the Laplace Transform; inverse transforms and translation theorems
2. Derivatives of a transform and convolution integrals

Learning Outcomes

1. Apply the Laplace transform method to solve linear engineering ODEs with initial conditions [Usage]
2. Model a damped mechanical oscillator or RLC circuit as a second-order linear ODE [Usage]
3. Analyze the stability and transient response of engineering systems described by systems of ODEs [Assessment]
4. Formulate state-space representations for multi-variable engineering dynamic systems [Usage]

7.31.4.3. Numerical Methods for ODEs and PDEs (36 hours) [Skills ABET-1] ↑ Back to top

Bibliography: (Boyce et al., 2017; Zill, 2018)

Topics

1. Phase portraits and linearization of nonlinear engineering systems
2. Stability of critical points and nonlinear systems

Learning Outcomes

1. Implement a fourth-order Runge-Kutta integrator for a mechanical vibration problem [Usage]
2. Determine the region of absolute stability for an explicit time-stepping scheme [Assessment]
3. Discretize the 2D heat equation using finite differences and set up the resulting linear system [Usage]
4. Compare finite difference and finite element approaches for solving an engineering PDE [Assessment]

7.31.5. Bibliography ↑ Back to top

Boyce, W. E., DiPrima, R. C., and Meade, D. B. (2017). Ecuaciones Diferenciales y Problemas con Valores en la Frontera. Limusa Wiley, 11th edition.

Zill, D. G. (2018). Ecuaciones Diferenciales con Aplicaciones de Modelado. Cengage Learning, 11th edition.