Ordinary differential equations
Solving equations that describe a function through its derivatives, by exact methods, models and numerical approximation.
An ordinary differential equation (ODE) describes a function through how it changes and the task is to reconstruct the functions that satisfy it. ODEs model motion, population change, cooling, mixing and oscillation. The chapter assumes the differentiation, integration and algebra of the earlier calculus material.
Lessons
- ODE fundamentals: order, linearity, solution families and the initial or boundary data that select one solution.
- First-order methods: separable, linear and exact equations and the substitutions that reduce other equations to these.
- Modelling with first-order equations: translating proportional change, relaxation and conservation laws into equations, then reading the solutions.
- Second-order constant-coefficient equations: characteristic roots as modes, forcing by undetermined coefficients, resonance and spring-mass systems.
- Existence, uniqueness and linear operators: the Lipschitz condition that guarantees a unique local solution and why every forced linear solution is one particular solution plus a homogeneous one.
- Numerical solutions: direction fields, Euler's method and Heun's method, with their error estimates. Read after the previous lesson, since the estimates rely on its Lipschitz condition.
- Choosing an ODE method: a checklist for classifying an equation and picking a method.
- Seen questions: past exam questions on exact equations, integrating factors and forced second-order equations.
Methods by structure
Each exact method relies on a structure the equation must have:
- Separable: separate the variables and integrate.
- First-order linear: an integrating factor turns the left side into a product-rule derivative.
- Exact: the equation is the total differential of a potential.
- Substitution: homogeneous and Bernoulli substitutions change variables into one of the cases above.
- Second-order constant-coefficient: an algebraic equation gives the exponential modes.
When no structure applies, a numerical method follows the slope field instead.
Where it leads
Euler's method steps along tangent lines. The next chapter, Taylor series, develops that idea into local polynomial models and measures the terms a single step leaves out.
Vocabulary checkpoints
- Order and degree: the order is the highest derivative present and the degree, when the equation is a polynomial in its derivatives, is the power of that derivative (ODE fundamentals).
- Solution formula and domain: a formula solves an ODE only on the interval or branch where it is defined and satisfies the equation (ODE fundamentals).
- Separable, linear and exact: these are independent tests, so one equation can pass several of them (first-order methods).
- Equilibrium and integration constant: an equilibrium is a constant solution, while a constant of integration labels a member of the solution family (modelling).
- Homogeneous and particular solutions: the homogeneous part carries the arbitrary constants and the particular part accounts for the forcing (second-order equations).
- Local and global error: the local error comes from one numerical step, while the global error also includes errors carried forward from earlier steps (numerical solutions).