Differential equations
Undergraduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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Topics typically covered
Standard syllabus
First-order equations
- Classification of ODEs; initial value problems and existence (informal)
- Direction fields and Euler's method for first-order equations
- Separable differential equations
- First-order linear equations; integrating factors
- Exact equations and integrating factors (where covered)
- Substitution methods and Bernoulli equations (optional)
- Modeling with first-order equations: growth/decay, mixing, and Newton's law of cooling
Second-order linear equations
- Second-order linear homogeneous equations with constant coefficients
- Characteristic equations; distinct, repeated, and complex roots
- Nonhomogeneous equations; method of undetermined coefficients
- Variation of parameters for second-order equations
Laplace transforms and systems
- Laplace transforms: definition, tables, and shifting theorems
- Solving IVPs with Laplace transforms and partial fractions
- Systems of first-order linear equations and matrix methods (introduction)
STEM / applied
Physical models
- Mechanical vibrations: spring-mass systems and resonance
- Electrical circuits: RLC circuits and analogies to mechanical systems
- Population and epidemic models (logistic, predator–prey overview)
- Forced oscillations and beats; amplitude and frequency response
Methods and technology
- Orthogonal trajectories and slope fields in applied settings
- Numerical methods: improved Euler and Runge–Kutta (introduction)
- Phase portraits for 2×2 linear systems
- Unit step and impulse functions via Laplace transforms (Heaviside)
- Engineering interpretation of solutions and parameters
Notes
Topics reflect common ordinary differential equations syllabi at US colleges and universities. PDEs, series solutions, and advanced transform methods are often taught in separate courses.