Chapter 7 ยท Introduction

Chapter 7 โ€” Introduction to Differential Equations

Laws that say how fast something changes โ€” and how calculus turns them into predictions about cooling coffee, growing populations and medicines in the bloodstream.

Pour a cup of coffee and leave it on the desk. Nobody hands you a formula for its temperature at 10:37. What physics does give you is a rule for how fast it cools: the rate of cooling is proportional to how much hotter the coffee is than the room. A bacterial colony obeys a similar rule โ€” the more bacteria there are, the faster new ones appear โ€” and so does a painkiller being cleared by the kidneys, a loan accumulating interest, or a parachutist slowing to a steady speed. In each case the law of nature is a statement about a derivative.

An equation that links an unknown function to its derivatives is a differential equation. Solving one means running from โ€œhow it changesโ€ back to โ€œwhat it isโ€ โ€” the same journey as antidifferentiation in Calculus I, but now the rate may depend on the unknown quantity itself. In this chapter you will learn to check solutions, to read the behaviour of solutions straight from a picture of slopes, to compute them step by step when no formula exists, and to solve the most important families exactly โ€” ending with the logistic equation that describes populations, epidemics and the spread of new technology.

Rates of changeโ†’Differential equations & IVPsโ†’Direction fieldsโ†’Equilibria & stabilityโ†’Euler's methodโ†’Separable equationsโ†’Logistic growth
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • state the order of a differential equation and verify a proposed solution by substitution;
  • distinguish general and particular solutions and solve initial-value problems, including motion problems and $y^\prime=ky$;
  • sketch solutions from a direction field, find equilibrium solutions and classify them as stable, unstable or semi-stable;
  • approximate solutions with Euler's method and explain how the error depends on the step size;
  • solve separable equations and use them to model cooling, growth and decay;
  • model limited growth with the logistic equation and interpret its carrying capacity.
Teaser โ€” The Rate Depends on the State

Watch a quantity change over time. The blue tangent line shows its rate of change at the moving point. Compare the last two readouts: the rate is always the same multiple of the current state โ€” that single rule is the differential equation, and it determines the whole curve.

time $t$โ€”
current stateโ€”
rate of changeโ€”
rate รท (state โˆ’ equilibrium)โ€”

Sections in this chapter

7.1

Basics of Differential Equations

Order, verifying solutions, general and particular solutions, initial-value problems, motion and $y^\prime=ky$.

$y^\prime=ky\iff y=Ce^{kt}$
Open section โ†’7.2

Direction Fields and Euler's Method

Slope fields, equilibria and their stability, and stepping along the slopes numerically.

$y_{k+1}=y_k+h\,f(x_k,y_k)$
Open section โ†’7.3

Separable Equations

Separate the variables and integrate both sides; exact solutions for growth and decay, cooling, mixing tanks and drug infusion.

$\displaystyle\int\frac{dy}{h(y)}=\int g(x)\,dx$
Open section โ†’7.4

The Logistic Equation

Growth that levels off at a carrying capacity: the S-curve and its inflection point, fitting to data, and harvesting.

$\dfrac{dP}{dt}=rP\left(1-\dfrac{P}{K}\right)$
Open section โ†’
โœ… Before you start
You will differentiate constantly to check solutions, so be fluent with the differentiation rules and the chain rule, and with derivatives as rates of change. Solving equations means integrating: revise antiderivatives and initial-value problems, substitution, integration by parts and, for Sections 7.3โ€“7.4, partial fractions. Exponential and logarithm laws (Calculus I ยง0.1) appear in almost every answer.
โ† 6.5 Work, Moments and Centres of Mass7.1 Basics of Differential Equations โ†’