Escalera de Matemáticas · Universidad

Ecuaciones Diferenciales

Ecuaciones sobre razones de cambio: métodos de primer orden, ecuaciones lineales de segundo orden, transformada de Laplace, sistemas, series y un primer vistazo a las EDP.

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49 temas en 7 unidades. Se apoya en Cálculo 2: Integrales y Series y Álgebra Lineal. Lleva a Análisis Numérico. ¿No estás seguro de que sea el curso adecuado? Encuentra tu peldaño.

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Unidad 1 First-Order Equations

7 temas

Classify equations and solve linear, exact and substitution-type first-order equations.

  1. Order, Linearity and Solutions
  2. Initial Value Problems, Existence and Uniqueness
  3. Linear Equations and Integrating Factors
  4. Exact Equations
  5. Homogeneous and Bernoulli Equations
  6. Autonomous Equations and the Phase Line
  7. Improved Euler and Runge–Kutta Methods

Unidad 2 First-Order Models

4 temas

Build and solve models of mixing, growth, cooling and motion.

  1. Mixing Problems
  2. Growth, Decay and Newton's Law of Cooling
  3. Population Models with Harvesting
  4. Falling Objects with Drag and RC Circuits

Unidad 3 Second-Order Linear Equations

12 temas

Solve constant-coefficient equations, homogeneous and forced, and model vibrations.

  1. Second-Order Linear Equations and Superposition
  2. Independence and the Wronskian
  3. Constant Coefficients: Distinct Real Roots
  4. Constant Coefficients: Complex Roots
  5. Constant Coefficients: Repeated Roots
  6. Reduction of Order
  7. Undetermined Coefficients
  8. Variation of Parameters
  9. Free Vibrations and Damping
  10. Forced Vibrations and Resonance
  11. RLC Circuits
  12. Higher-Order Linear Equations

Unidad 4 Laplace Transforms

7 temas

Turn differential equations into algebra, including steps and impulses.

  1. The Laplace Transform
  2. Transforms of Derivatives and Shifts
  3. Inverse Transforms
  4. Solving Initial Value Problems with Laplace
  5. Step Functions and Piecewise Forcing
  6. Impulses and the Delta Function
  7. Convolution

Unidad 5 Systems of Differential Equations

9 temas

Solve linear systems with eigenvalues and read phase portraits, then study nonlinear systems.

  1. Systems in Matrix Form
  2. Solving Systems with Real Eigenvalues
  3. Phase Portraits
  4. Systems with Complex Eigenvalues
  5. Systems with Repeated Eigenvalues
  6. The Matrix Exponential
  7. Nonhomogeneous Systems
  8. Nonlinear Systems and Linearization
  9. Predator–Prey and the Pendulum

Unidad 6 Series Solutions

3 temas

Solve equations with power series near ordinary and singular points.

  1. Power Series Solutions near Ordinary Points
  2. Cauchy–Euler Equations
  3. Regular Singular Points and Frobenius

Unidad 7 Fourier Series and PDEs

7 temas

Expand functions in Fourier series and solve the heat, wave and Laplace equations.

  1. Boundary Value Problems and Eigenfunctions
  2. Fourier Series
  3. Fourier Sine and Cosine Series and Convergence
  4. Separation of Variables
  5. The Heat Equation
  6. The Wave Equation
  7. Laplace's Equation