# Part IA - Differential Equations

## Lectured by M. G. Worster, Michaelmas 2014

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

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# Contents

- V Full version
- 0 Introduction
- 1 Differentiation
- 1.1 Differentiation
- 1.2 Small o and big O notations
- 1.3 Methods of differentiation
- 1.4 Taylor's theorem
- 1.5 L'Hopital's rule
- 2 Integration
- 3 Partial differentiation
- 3.1 Partial differentiation
- 3.2 Chain rule
- 3.3 Implicit differentiation
- 3.4 Differentiation of an integral wrt parameter in the integrand
- 4 First-order differential equations
- 4.1 The exponential function
- 4.2 Homogeneous linear ordinary differential equations
- 4.3 Forced (inhomogeneous) equations
- 4.4 Non-constant coefficients
- 4.5 Non-linear equations
- 4.6 Solution curves (trajectories)
- 4.7 Fixed (equilibrium) points and stability
- 4.8 Discrete equations (Difference equations)
- 5 Second-order differential equations
- 5.1 Constant coefficients
- 5.2 Particular integrals
- 5.3 Linear equidimensional equations
- 5.4 Difference equations
- 5.5 Transients and damping
- 5.6 Impulses and point forces
- 5.7 Heaviside step function
- 6 Series solutions
- 7 Directional derivative
- 7.1 Gradient vector
- 7.2 Stationary points
- 7.3 Taylor series for multi-variable functions
- 7.4 Classification of stationary points
- 7.5 Contours of f(x, y)
- 8 Systems of differential equations
- 9 Partial differential equations (PDEs)