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Differential Equations
Introduction
Basic Concepts
Practice: Order and Degree
General and Particular Solutions of a Differential Equation
Practice: Verifying Solutions
Methods of Solving First Order, First Degree Differential Equations
Practice: Variables Separable
Practice: Homogeneous Equations
Practice: Linear Differential Equations
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
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Differential EquationsIntroduction

Reading time: ~15 min

Every evening Riya's mother sets dahi for the next day. She warms the milk, stirs in one spoonful of yesterday's curd (the jaman), covers the bowl and leaves it in a warm corner. By morning the whole bowl has set.

That spoonful carries about one lakh bacteria, and in warm milk each of them splits in two roughly once an hour, so the colony doubles every hour. Nobody hands the bacteria a formula. The only rule they follow is about speed: the more of them there are, the faster their number grows.

The green curve is the number of bacteria N (in lakh) after t hours, and the orange dash is its tangent right now: its slope is how fast the colony is growing. Drag the time and complete all three missions. Keep an eye on the last number in the readout.

t = ${t} hours

The colony N = 2^t lakh. The steeper the tangent, the faster it grows.

At 2 lakh the colony was growing by about 1.39 lakh per hour, at 4 lakh by 2.77, at 8 lakh by 5.55. Each time the colony doubled, its speed of growth , and the ratio "rate ÷ N" never moved from about .

So the dahi obeys one simple rule: rate of growth = 0.69 × size. Written with a derivative,

\dfrac{dN}{dt} = 0.69\,N

This is an equation, but its unknown is not a number. The unknown is a whole function N(t), and the equation only tells us about its rate of change. An equation like this is called a differential equation. Reading and solving them is what this chapter is about.

Equations That Talk About Change

In Class XI and in Chapter 5 you learnt to go one way: given a function, find its derivative. Given y = x^2, you write \frac{dy}{dx} = 2x at once. This chapter goes the other way. We are handed a relation that involves a derivative, such as

x\dfrac{dy}{dx} = y \qquad \ldots (1)

and asked: which functions y = f(x) make it true?

Try y = 3x. Then \frac{dy}{dx} = 3, and x \cdot 3 = 3x, which is y again. So y = 3x equation (1). Now try y = x^2: the left side is x \cdot 2x = 2x^2, which is y.

Equations like (1) are the everyday language of science. Physics uses them for motion, cooling and electric circuits, chemistry for the speed of reactions, biology for populations like our dahi colony, and geology, anthropology and economics for growth and decay of all kinds. Solving one means predicting: from a rule about change, we recover the whole story.

The name itself is old. Leibniz first wrote aequatio differentialis, "differential equation", in 1676. Within about twenty years he and the Bernoulli brothers, Jacob and Johann, had found the methods for the three kinds of equations you will solve in this chapter: equations with separable variables, homogeneous equations and linear equations.

In this chapter you will learn what makes an equation a differential equation, how to read its order and degree, what its general and particular solutions look like, and three methods for solving first-order equations. On the way you will solve the dahi equation exactly.

Sina