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Differential Equations

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Author(s): M. G. Worster, ed. Dexter Chua
Series: Cambridge Mathematical Tripos Part IA Lecture Notes
Publisher: University of Cambridge
Year: 2014

Language: English
City: Cambridge
Tags: maths; mathematics; math; advanced; college; university; higher; further; pure; applied; calculus; analysis; differential; integral; single variable; multiple variables; multivariable; vector; tensor; one variable; many variables

Introduction
Differentiation
Differentiation
Small o and big O notations
Methods of differentiation
Taylor's theorem
L'Hopital's rule
Integration
Integration
Methods of integration
Partial differentiation
Partial differentiation
Chain rule
Implicit differentiation
Differentiation of an integral wrt parameter in the integrand
First-order differential equations
The exponential function
Homogeneous linear ordinary differential equations
Forced (inhomogeneous) equations
Constant forcing
Eigenfunction forcing
Non-constant coefficients
Non-linear equations
Separable equations
Exact equations
Solution curves (trajectories)
Fixed (equilibrium) points and stability
Perturbation analysis
Autonomous systems
Logistic Equation
Discrete equations (Difference equations)
Second-order differential equations
Constant coefficients
Complementary functions
Second complementary function
Phase space
Particular integrals
Guessing
Resonance
Variation of parameters
Linear equidimensional equations
Difference equations
Transients and damping
Impulses and point forces
Dirac delta function
Heaviside step function
Series solutions
Directional derivative
Gradient vector
Stationary points
Taylor series for multi-variable functions
Classification of stationary points
Contours of f(x, y)
Systems of differential equations
Linear equations
Nonlinear dynamical systems
Partial differential equations (PDEs)
First-order wave equation
Second-order wave equation
The diffusion equation