Math Ladder · College

Differential Equations

Equations about rates of change: first-order methods, second-order linear equations, Laplace transforms, systems, series and a first look at PDEs.

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49 topics in 7 units. Builds on Calculus 2: Integrals and Series and Linear Algebra. Leads to Numerical Analysis. Not sure this is the right course? Find your rung.

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Unit 1 First-Order Equations

7 topics

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

Unit 2 First-Order Models

4 topics

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

Unit 3 Second-Order Linear Equations

12 topics

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

Unit 4 Laplace Transforms

7 topics

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

Unit 5 Systems of Differential Equations

9 topics

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

Unit 6 Series Solutions

3 topics

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

Unit 7 Fourier Series and PDEs

7 topics

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
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