# Scaling of Differential Equations¶

Contents:

- Scaling of Differential Equations
- Preface
- Dimensions and units
- Ordinary differential equation models
- Exponential decay problems
- Fundamental ideas of scaling
- The basic model problem
- The technical steps of the scaling procedure
- Making software for utilizing the scaled model
- Scaling a generalized problem
- Variable coefficients
- Scaling a cooling problem with constant temperature in the surroundings
- Scaling a cooling problem with time-dependent surroundings
- Scaling a nonlinear ODE
- SIR ODE system for spreading of diseases
- SIRV model with finite immunity
- Michaelis-Menten kinetics for biochemical reactions

- Vibration problems
- Exercises
- Exercise 2.1: Perform unit conversion
- Problem 2.2: Scale a simple formula
- Exercise 2.3: Perform alternative scalings
- Problem 2.4: A nonlinear ODE for vertical motion with air resistance
- Exercise 2.5: Solve a decay ODE with discontinuous coefficient
- Exercise 2.6: Implement a scaled model for cooling
- Problem 2.7: Decay ODE with discontinuous coefficients
- Exercise 2.8: Alternative scalings of a cooling model
- Exercise 2.9: Projectile motion
- Problem 2.10: A predator-prey model
- Problem 2.11: A model for competing species
- Problem 2.12: Find the period of sinusoidal signals
- Problem 2.13: Oscillating mass with sliding friction
- Problem 2.14: Pendulum equations
- Exercise 2.15: ODEs for a binary star
- Problem 2.16: Duffing’s equation
- Problem 2.17: Vertical motion in a varying gravity field
- Problem 2.18: A simplified Schroedinger equation

- Exponential decay problems
- Basic partial differential equation models
- Advanced partial differential equation models