In the intro to Chapter 9 they note that we’ve already seen differential equations in Chapter 3, p. 223, and on p.312, and in Chapter 4 (see p 405).  So I’ll start there.

• Section 3.4: First encounter: in “Some Applications of the Exponential Function”, y’ = ay is introduced.  “Since Eq. (8) expresses a relation between the function and its derivative, it is called the differential equation of the exponential function”.  They show that the exponential function is the unique solution (this argument is worthwhile, actually, because it is a small proof).
• Section 3.4: more y’ = ay.  Examples associated with the exponential function: compound interest, radioactive decay, Newton’s law of cooling, atmospheric pressure with height above the surface of the Earth,  the law of mass action (chemical reactions), switching on and off an electric circuit.  Newton’s law of cooling,  the law of mass action, and the electric circuit involve differential equations.
• Section 3.16a: Differential equations of trigonometric functions.  In 3.16a they intro diff eqs.  Diff eqs move beyond equations y’ = f(x) to “more general relationships between y and derivatives of y”.
• Section 3.16b: Define sine and cosine via a differential equation (u” + u = 0) and an initial condition.  “Any function u = F(x) satisfying the equation, …, is a solution.”  They then show that shifts of solutions are solutions and linear combos are solutions, and scalar multiples are solutions, so the properties of linearity.  Initial conditions single out a specific solution.  They also derive cos(x+y) = cos x cos y – sin x sin y using the differential equation.  They also note that pi/2 can then be defined via “the smallest positive value of x for which cos x = 0.”
• Section 4.4a: Newton’s law of motion, a relationship “from which we hope to determine the motion”.  They define diff eq and solution again.
• Section 4.4b/c: Motion of falling bodies and motion constrained to a curve.
• Section 4.5: free fall of a body in the air (find terminal velocity under this model)
• Section 4.6: simplest elastic vibration: motion of a spring.
• Section 4.7abcde: motion on a given curve.  The differential equation and its solution.  Particle sliding down a curve.  Discussion of the motion.  The ordinary pendulum.  The cycloidal pendulum.
• Section 4.8abc: Motion in a gravitational field.  Newton’s universal law of gravitation.  Circular motion about the center of attraction.  Radial motion – escape velocity.
• Chapter 9:  They summarize the differential differential equations that have been encountered above.  This chapter is differential equations for the simplest types of vibration.
• 9.1ab: Vibration problems of mechanics and physics.  The simplest mechanical vibrations (forced second order constant coefficient equation).  Electrical oscillations (similar).
• 9.2abc: Solution of the homogeneous equation.  Free oscillations. a:The formal solution.  b:Interpretation of the solution.  c: Fulfilment of given initial conditions. Uniqueness of the solution. In the formal solution they construct the characteristic equation and distinguish the three cases of roots.  For the solutions in complex form they introduce Euler’s formula.  In interpretation they introduce “damping”, “damped harmonic oscillations”, “attenuation constant”, “natural frequency”.
• 9.3abcde: The nonhomogeneous equation.  Forced oscillations.  a: general remards.  Superposition.  b: Solution of the nonhomogeneous equation.  c: The resonance curve.  d: Further discussion of the oscillation.  e: Remarks on the construction of recording instruments.