1 Introduction

1.1 Deriving a differential equation in a simple physical model

1.2 Graphical solution of model ([eqn:ch1_grav_model_with_air_res])

1.3 Guessing solutions

1.4 Numerical solution

1.5 Existence and uniqueness theorems

1.6 Overview of course

2 Solutions to ODEs

2.1 Verifying solutions

2.2 Formal definition

2.3 Method of undetermined coefficients

2.4 Alternative notation

3 First order equations

3.1 Tangent fields

3.2 First order linear autonomous equation

3.3 Explicit and implicit ODEs

3.4 Interlude - Leibniz rule for differentiating under the integral sign

3.5 Explicit first order linear non-autonomous equations

3.6 Separable first order equations

3.7 Failure of global existence

3.8 Failure of local uniqueness

3.9 Implicit solutions

3.10 Exact equations

3.11 Integrating factors

4 Series solutions

5 Higher order linear autonomous equations

5.1 Second order

5.2 Idealized spring-mass system - harmonic oscillator

5.3 \(k\ge3\)-th order linear autonomous equations

6 Existence and uniqueness

6.1 Uniqueness - first order

6.2 Uniqueness - systems of ODEs and higher order ODEs

6.3 Existence - first order

6.4 Existence - higher order ODEs

7 Systems of first order ODEs

7.1 Direction fields for autonomous planar systems

7.2 Planar phase portraits

7.3 Phase portraits and direction fields for second order autonomous ODEs

7.4 Linear autonomous planar ODEs

7.5 Phase portraits of non-linear planar ODEs

7.6 The exponential of a matrix

7.7 Non-autonomous linear system

8 Higher order linear equations

8.1 Existence and uniqueness

8.2 The homogeneous equation

8.3 The non-homogeneous equation - particular solutions

8.4 Forced spring

9 Numerical solution I: first steps

9.1 Euler’s method

9.2 Euler’s method for higher order equations

9.3 Trapezoidal rule method

9.4 Fixed-point equations

10 Numerics II: Multistep methods

10.1 Adams-Bashforth methods

10.2 General linear multistep methods

10.3 Interlude - difference equations

10.4 Global convergence

10.5 Deriving methods algebraically

10.6 Adams-Moulton methods

10.7 Backward differentiation formulas

10.8 Dahlquist barriers

10.9 Summary

11 Numerics III: Runge-Kutta methods

11.1 Gaussian quadrature

11.2 Explicit Runge-Kutta (ERK) schemes

11.3 Implicit Runge-Kutta (IRK) methods

11.4 Collocation methods

11.5 Global convergence

12 Numerics IV: Stiff ODEs

12.1 An example of a stiff ODE

12.2 Linear stability domain and \(A\)-stability

12.3 Linear stability domain of Runge-Kutta methods

12.4 Linear stability domain of multistep methods

12.5 Fixed-point iteration and stability

13 Numerics V: Error control

13.1 Embedded Runge-Kutta methods

13.2 Milne device