1 First-Order Equations and the Qualitative View
A differential equation relates an unknown function to its derivatives. Such equations are the language in which the laws of mechanics, electricity, chemical kinetics, population growth, epidemics and economic dynamics are written, because each of these laws says how a state changes, not what it is. Newton’s second law gives the acceleration of a body in terms of the forces on it; the law of mass action gives the rate of a reaction in terms of the concentrations; a model of an epidemic gives the rate of new infections in terms of the numbers of susceptible and infected people. The state at later times must be deduced.
This course is about how that deduction is done, and about what can be known when it cannot be done in closed form, which is the usual case. It begins with first-order equations, where explicit methods are most successful, and with the qualitative viewpoint that will carry the rest of the course: understanding the behaviour of solutions without computing them.
1.2 Linear first-order equations
1.3 Separable equations
1.4 Exact equations and integrating factors
1.5 Equations reducible to linear form
1.6 Autonomous equations and the phase line
1.7 First-order equations in software
1.8 Practice Problems
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