MU-205 Ordinary Differential Equations

Unit · year 2

MU-205 · Ordinary Differential Equations

Threads change24 lectures5 theorems

Existence, structure and solution of differential equations.

PREREQUISITES

L01	What an ODE Is: Order, Linearity, Solutions —
L02	First-Order Separable Equations —
L03	Linear First-Order Equations and Integrating Factors —
L04	Exact Equations —
L05	Existence and Uniqueness: The Question The Picard–Lindelöf theorem
L06	The Picard–Lindelöf Theorem The Picard–Lindelöf theorem
L07	Picard Iteration in Practice The Picard–Lindelöf theorem
L08	Continuation and Blow-Up of Solutions The Picard–Lindelöf theorem
L09	Second-Order Linear Equations —
L10	Superposition and the Solution Space Superposition for linear ODEs
L11	The Wronskian and Linear Independence The Wronskian and independence
L12	Abel's Identity The Wronskian and independence
L13	Constant-Coefficient Equations Superposition for linear ODEs
L14	Reduction of Order —
L15	The Method of Undetermined Coefficients —
L16	Variation of Parameters Variation of parameters
L17	Forced Oscillation and Resonance Variation of parameters
L18	Series Solutions and Regular Singular Points —
L19	Systems of First-Order Equations —
L20	The Matrix Exponential —
L21	Phase Portraits for Linear Systems Linear stability and the phase plane
L22	Classification of Equilibria Linear stability and the phase plane
L23	Nonlinear Systems and Linearisation Linear stability and the phase plane
L24	Synthesis: Local Theory, Global Behaviour The Picard–Lindelöf theorem, Linear stability and the phase plane

A Lipschitz ODE has a unique local solution.

Solutions of a linear equation form a vector space.

A non-vanishing Wronskian certifies linear independence of solutions.

A particular solution from the homogeneous solutions.

Eigenvalues of the linearisation classify equilibria.