Chapter 9 · Introduction

Chapter 9 — Differential Equations

Nature rarely hands us a formula. It hands us a rule for how something changes — how fast tea cools, how fast a drug leaves the blood, how fast a city grows. A differential equation is that rule; solving it recovers the formula.

Pour a cup of chai at $90^\circ\text{C}$ and leave it on the table in a $25^\circ\text{C}$ room. Nobody can tell you in advance a formula for its temperature at every minute — but a physicist can tell you a rule: the tea loses heat at a rate proportional to how much hotter it is than the room. When the tea is very hot it cools fast; as it approaches room temperature the cooling slows almost to nothing. The same kind of rule describes a dose of paracetamol leaving the bloodstream (the rate of removal is proportional to the amount present), a bank balance under continuous compounding, the number of bacteria in a culture, and the charge on a capacitor in a phone charger. In every case we know how a quantity changes — its derivative — and we want the quantity itself.

An equation that involves an unknown function together with its derivatives is a differential equation. It is the language in which almost every law of physics, chemistry, biology and economics is written. In this chapter you learn to read such an equation — its order and degree — and to tell the general solution (a whole family of curves) from the particular solution picked out by a starting value. Then you meet the three classes of first-order equations that the Board and JEE expect you to solve by hand: equations with variables separable, homogeneous equations (solved by the substitution $y=vx$), and linear equations $\dfrac{dy}{dx}+Py=Q$ (solved with an integrating factor). Every method ends with an integration, so this chapter is where the integrals of Chapter 7 — Integrals finally start predicting the future. Throughout, $\log x$ means the natural logarithm $\log_e x$.

Derivatives (Ch 5)→Integrals (Ch 7)→Order, degree, solutions→Variables separable→Homogeneous: y = vx→Linear: integrating factor→Growth, decay, cooling, mixing

🎯 By the end of this chapter you can…

recognise a differential equation, find its order, and find its degree when it is defined (or explain why it is not);
verify that a given function — explicit or implicit — is a solution, and distinguish a general solution with arbitrary constants from a particular solution fixed by initial conditions;
read a slope field and sketch the solution curve through a given point;
solve equations with variables separable, and use them to model exponential growth and decay, half-life, doubling time and Newton's law of cooling;
test whether a first-order equation is homogeneous and solve it with the substitution $y=vx$ (or $x=vy$);
solve linear equations $\dfrac{dy}{dx}+Py=Q$ and $\dfrac{dx}{dy}+P_1x=Q_1$ with an integrating factor, including mixing and circuit problems;
answer the Board's 1-, 2-, 3- and 5-mark questions on this chapter and the JEE Main / JEE Advanced problems built on it.

The short grey dashes form a slope field: at each point $(t,T)$ the dash has the slope $\dfrac{dT}{dt}=-k(T-T_s)$ prescribed by the cooling law, with $k=0.08$ per minute (an illustrative value) and room temperature $T_s$. Play lets time run for a cup of chai at $90^\circ\text{C}$ and a cold drink at $5^\circ\text{C}$. Tap or click anywhere to start another drink at that time and temperature. Every curve follows the dashes — and every one is drawn towards the room temperature.

room $T_s$:

time $t$—

chai $T$—

cold drink $T$—

chai cooling rate $\tfrac{dT}{dt}$—

Sections in this chapter

9.1

Order, Degree and Solutions of a Differential Equation

What a differential equation is; its order and degree; general and particular solutions; slope fields and initial conditions.

$\dfrac{d^2y}{dx^2}+y=0$: order $2$, degree $1$

Open section →9.2

Equations with Variables Separable

Put all the $y$'s with $dy$ and all the $x$'s with $dx$, then integrate both sides; growth, decay, half-life and cooling.

$\displaystyle\int\frac{dy}{h(y)}=\int g(x)\,dx+C$

Open section →9.3

Homogeneous Differential Equations

Equations $\dfrac{dy}{dx}=g\!\left(\dfrac yx\right)$ whose slopes depend only on the direction from the origin; the substitution $y=vx$.

$y=vx:\ \ x\dfrac{dv}{dx}=g(v)-v$

Open section →9.4

Linear Differential Equations

$\dfrac{dy}{dx}+Py=Q$: multiply by the integrating factor to make the left side an exact derivative; mixing tanks and circuits.

$y\cdot\text{I.F.}=\displaystyle\int Q\cdot\text{I.F.}\,dx+C$

Open section →

✅ Before you start

Solving a differential equation means undoing a derivative, so you need both directions fluently. From Chapter 5 of this course: the chain rule and implicit differentiation (Section 5.2), the derivatives of $e^x$ and $\log x$ (Section 5.3), and second-order derivatives (Section 5.4). From Chapter 7 — Integrals: the standard integrals, integration by substitution, partial fractions, and integration by parts, including $\displaystyle\int e^x\big[f(x)+f'(x)\big]dx=e^xf(x)+C$. You will also use the laws of logarithms and exponentials constantly ($e^{a+b}=e^ae^b$, $\log|xy|=\log|x|+\log|y|$). Quick self-test: can you find $\displaystyle\int\frac{dx}{1+x^2}$ and $\dfrac{d}{dx}\left(e^{x^2}\right)$ without looking anything up? (Answers: $\tan^{-1}x+C$ and $2xe^{x^2}$.)

← 8.2 Areas Bounded by Lines, Parabolas, Circles and Ellipses 9.1 Order, Degree and Solutions of a Differential Equation →