Fast Track to Differential Equations: Applications-Oriented – Comprehensible – Compact

Preface
	Words of Thanks
Preface to the English Edition
Contents
1 Prerequisites from Calculus
	1.1 Exponential and Logarithm Functions
		1.1.1 Exponential Function exp(x) as a Power Series
		1.1.2 Properties of exp(x)
		1.1.3 Exponential Function
		1.1.4 Change of Base and Hyperbolic Functions
		1.1.5 Logarithm Function
	1.2 Integral Calculus
		1.2.1 Definite Integral
		1.2.2 Fundamental Theorem of Integral Calculus
		1.2.3 Computing Integrals
	1.3 Applications of Integral and Differential Calculus
		1.3.1 Escape Velocity
		1.3.2 Elasticity in Economics
		1.3.3 Harmonic Sum and Harmonic Series
		1.3.4 Optimal Stopping
		1.3.5 Fermat's Principle, Snell's Law of Refraction
	1.4 Parametrized Curves or Vector-Valued Functions
		1.4.1 Definition and Examples
		1.4.2 Ellipse
		1.4.3 Tangents of a Curve, Vectors of Velocity
	1.5 Exercises
2 First Order Differential Equations
	2.1 Concepts and Definitions
	2.2 Geometrical Aspects, Direction Fields, Isoclines
	2.3 Linear First Order Differential Equations
		2.3.1 Homogeneous Case
		2.3.2 Nonhomogeneous Case
	2.4 Existence and Uniqueness of Solutions
	2.5 Separable Differential Equations
	2.6 Autonomous Differential Equation and Stability
	2.7 Exercises
3 First Order Applications
	3.1 Population Model
	3.2 Newton's Law of Cooling
	3.3 Radioactivity and Radiocarbon Dating
		3.3.1 Radioactivity
		3.3.2 Radiocarbon Dating
		3.3.3 More on Radiocarbon Dating
	3.4 Compound Interest
	3.5 Specified Elasticity Function in Economics
	3.6 Evaporation of a Rain Drop
	3.7 Mixing Problem
	3.8 Vertical Launch of a Rocket Without Air Drag
	3.9 Gravitational Funnel
	3.10 Nuclear Waste Disposal
	3.11 Barometric Formulas
		3.11.1 Isothermal Model
		3.11.2 Model for Linear Temperature Decrease
		3.11.3 General Model
	3.12 Liquid Container
	3.13 Electric Circuit
	3.14 Catenary
	3.15 Global Warming
	3.16 Brachistochrone Problem
	3.17 Cosmology
		3.17.1 Historical Facts
		3.17.2 Extent and Age of the Universe
		3.17.3 Olbers' Paradox
		3.17.4 Friedmann-Lemaître Equation
		3.17.5 Einstein-De Sitter Model
		3.17.6 Model for a Universe with Positive Space Curvature
	3.18 Orthogonal Trajectories
	3.19 Exercises
4 Second Order Differential Equations and Systems with Applications
	4.1 Second Order Differential Equation
	4.2 System of First Order Differential Equations
	4.3 Projectile Motion
	4.4 Modeling with Air Drag
	4.5 Coriolis Force in Meteorology
	4.6 Vector Fields and Field Lines
	4.7 Helmholtz Coils
	4.8 Oscillations and Resonance
	4.9 Curve of Pursuit
	4.10 Coupled Pendulums
	4.11 Predator-Prey Problem
	4.12 Periodic Solutions and Limit Cycles
	4.13 Two-Body Problem in Celestial Mechanics
		4.13.1 Historical Remarks
		4.13.2 Proof of Kepler's Second Law
		4.13.3 Proof of Kepler's First Law
		4.13.4 Proof of Kepler's Third Law
	4.14 Exercises
5 Numerical Methods with Applications
	5.1 Basics
	5.2 Euler's Method
	5.3 Error Considerations
	5.4 Heun's Method
	5.5 Runge-Kutta Method
	5.6 Numerical Considerations on Systems of Differential Equations
	5.7 Trajectories of Tennis Balls
		5.7.1 Modeling
		5.7.2 Data and Comparison of Forces
		5.7.3 Mathematica Program for Computations and Diagrams
		5.7.4 Topspin Trajectories
		5.7.5 Slice Trajectories
		5.7.6 Comparing Topspin Trajectory with Trajectory  in a Vacuum
	5.8 Mathematical Model for Skydiving
		5.8.1 Analytical Model
		5.8.2 Numerical Model
	5.9 Near-Earth Satellite Orbits
	5.10 Climate Change
	5.11 Brownian Motion and Langevin Equation
	5.12 Kalman Filter
		5.12.1 Theory
		5.12.2 Simulation of an Example
	5.13 Exercises
6 Solutions
Appendix  References
Index