Differential Equations II

Year 2 · 6 sections · 60 concepts · 146 practice questions

Start free (beta)Try chapter 1 free, no sign-up →

What's covered

free1 · Linear systems11 concepts

Overview 1 · The plan of this section
Definition 1.1 · Linear systems and the matrix exponential
Proposition 1.2 · Properties of the matrix exponential
Theorem 1.3 · Solution of the constant-coefficient system
Definition 1.4 · Fundamental matrix and Wronskian
Proposition 1.5 · Liouville's formula for the Wronskian
Proposition 1.6 · Computing e^{tA} from eigenvalues
Example 1.7 · A system with a repeated eigenvalue
Theorem 1.8 · Variation of parameters for systems
Proposition 1.9 · Higher-order linear equations as systems
Check 1.10 · Check: solving linear systems

2 · Phase plane10 concepts

Overview 2 · The plan of this section
Definition 2.1 · Autonomous systems, orbits and fixed points
Theorem 2.2 · Classification of planar linear fixed points
Definition 2.3 · Linearisation and hyperbolic fixed points
Theorem 2.4 · Hartman–Grobman theorem
Definition 2.5 · Nullclines
Definition 2.6 · Conserved quantities
Proposition 2.7 · Strict minima of a conserved quantity are centres
Example 2.8 · The pendulum phase portrait
Check 2.9 · Check: sketching a phase portrait

3 · Stability and Lyapunov9 concepts

Overview 3 · The plan of this section
Definition 3.1 · Stability of a fixed point
Theorem 3.2 · Linear stability test
Definition 3.3 · Lyapunov function
Theorem 3.4 · Lyapunov's stability theorem
Definition 3.5 · Periodic orbits and limit cycles
Theorem 3.6 · Poincaré–Bendixson theorem
Proposition 3.7 · Bendixson–Dulac criterion
Check 3.8 · Check: deciding stability

4 · Series solutions10 concepts

Overview 4 · The plan of this section
Definition 4.1 · Ordinary and singular points
Theorem 4.2 · Series solutions at an ordinary point
Example 4.3 · Airy's equation
Example 4.4 · Legendre's equation and Legendre polynomials
Definition 4.5 · Regular singular points
Proposition 4.6 · The indicial equation
Theorem 4.7 · Frobenius method
Example 4.8 · Bessel functions
Check 4.9 · Check: series solutions

5 · Existence and uniqueness10 concepts

Overview 5 · The plan of this section
Definition 5.1 · Lipschitz conditions
Proposition 5.2 · C¹ functions are locally Lipschitz
Theorem 5.3 · Picard's theorem for systems
Lemma 5.4 · Grönwall's inequality
Corollary 5.5 · Uniqueness of solutions
Theorem 5.6 · Continuous dependence on initial data
Example 5.7 · Finite-time blow-up
Example 5.8 · Non-uniqueness without a Lipschitz condition
Check 5.9 · Check: is this IVP well posed?

6 · Boundary value problems10 concepts

Overview 6 · The plan of this section
Definition 6.1 · Boundary value problems
Example 6.2 · Eigenvalues of y'' + λy = 0 with Dirichlet conditions
Definition 6.3 · Sturm–Liouville problems
Proposition 6.4 · Sturm–Liouville operators are self-adjoint
Theorem 6.5 · Eigenvalues are real and eigenfunctions orthogonal
Theorem 6.6 · Eigenfunction expansions
Example 6.7 · Solving a forced BVP by expansion
Proposition 6.8 · Green's function
Check 6.9 · Check: boundary value problems

Try it: a free card from this module

This is exactly what studying looks like inside UniMaths.

Proposition 1.2 · Properties of the matrix exponential

$\frac{\mathrm d}{\mathrm dt}e^{tA} = Ae^{tA}$ ; $e^{A+B} = e^Ae^B$ only if $AB = BA$ ; conjugation passes through; $\det e^{tA} = e^{t\operatorname{tr}A}$ .

Proof

Learn all 60 concepts like this

Cards, step-by-step proofs, and 146 questions scheduled so you review right before you forget.

Start free (beta)Try chapter 1 first