DP Maths AI HL 5.14 Differential Equations

DP Maths AI HL 5.14 Differential Equations - Full notes

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DP Maths AI HL | 5.14 Differential Equations

DP MATHEMATICS AI · TOPIC 5: CALCULUS

5.14

Differential Equations

Full notes for Applications and Interpretation HL

Contents

1	What is a differential equation?
2	Forming differential equations from context
3	Verifying solutions
4	General and particular solutions
5	Solving dy/dx = f(x) by integration
6	Exponential growth: dy/dt = ky
7	Exponential decay and half-life
8	Newton's law of cooling
9	Separation of variables
10	Slope fields: construction
11	Slope fields: interpretation
12	Euler's method
13	Euler's method with technology and step size
14	Equilibrium solutions and long-term behaviour
15	The logistic growth model
16	Choosing and evaluating a model
17	Technology for differential equations
18	Mixed review: worked problems
19	Coupled systems and phase portraits (HL)
20	Second-order differential equations (HL)
21	Investigation: modelling the spread of a rumour
22	Exam-style questions with markscheme
23	Common misconceptions and links
24	Check your understanding
25	Summary and topic map
26	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the practice exercise.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
form differential equations from descriptions of rates
verify that a function satisfies a differential equation
find general and particular solutions
solve dy/dx = f(x) and dy/dt = ky
use separation of variables for simple equations
sketch and interpret slope fields
use Euler's method and judge its accuracy
interpret equilibrium and logistic models

HL booklet: this booklet contains all the SL content plus coupled systems with phase portraits and second-order equations.

Guiding questions

How can a rule about rates of change predict the future of a system? How do we solve equations whose unknown is a function?

How this topic is examined

Differential equations connect calculus with modelling. Expect questions that form a DE from a description, solve dy/dt = ky or separable equations, sketch solution curves on slope fields, use Euler's method with a table or technology, and interpret long-term behaviour.

Show the separation and integration steps, include the constant immediately, and interpret solutions in context with units.

Mathematics: Applications and Interpretation places strong emphasis on technology, modelling and interpretation. Most papers allow a GDC, so you are expected to use it efficiently, record the method you used, and interpret every answer in context.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working, including what you entered into a GDC
Show that	obtain the given result; every step must be shown and the result must not be assumed
Estimate	obtain an approximate value
Sketch	a diagram showing the general shape and key features, labelled
Interpret / Comment	explain the meaning of a result in context, or give a judgement
Justify / Explain	give valid reasons or evidence
Hence	use the previous result

How to use these notes

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or explanation of a key result
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given), graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet

1 What is a differential equation?

An equation involving a derivative.

Think about it

A population grows faster when it is larger. How could you write that as an equation about dP/dt?

A differential equation (DE) relates an unknown function to its derivative(s), for example dy/dx = 3x2 or dP/dt = 0.04P. Solving it means finding the function. DEs describe how systems change: populations, temperatures, money in accounts, drug concentrations, the motion of objects.

2 Forming differential equations from context

Translate words into rates.

Description	Differential equation
P grows at a rate proportional to P	dP/dt = kP
the temperature difference decreases at a rate proportional to it	dT/dt = −k(T − Ts)
volume decreases at a constant 5 L/min	dV/dt = −5
rate is proportional to the square root of h	dh/dt = k√h
growth limited by a capacity K	dP/dt = kP(1 − P/K)

Practice exercise 1

1. Write a DE: the rate of change of the amount A of a drug in the blood is proportional to A, and A decreases.

2. Write a DE: the number N of people who have heard a rumour grows at a rate proportional to the number who have not heard it, in a town of 5000.

3 Verifying solutions

Substitute and check.

WORKED EXAMPLE 1 Verifying (standard)

Show that y = 3e2x satisfies dy/dx = 2y.

Solution

dy/dx = 6e2xA1

2y = 6e2x, so dy/dx = 2yAG

Practice exercise 2

1. Verify that y = x2 + 5 satisfies dy/dx = 2x.

2. Verify that T = 20 + 60e−0.1t satisfies dT/dt = −0.1(T − 20).

4 General and particular solutions

The arbitrary constant.

A general solution contains an arbitrary constant (a family of curves). An initial condition, such as y(0) = 4, selects one curve: the particular solution.

Figure 1. The family of solutions of dy/dx = 2x.

5 Solving dy/dx = f(x) by integration

Integrate both sides.

WORKED EXAMPLE 2 Direct integration (standard)

Solve dy/dx = 6x2 − 4 with y(1) = 3.

Solution

y = 2x3 − 4x + cA1

3 = 2 − 4 + c: c = 5M1

y = 2x3 − 4x + 5A1

Practice exercise 3

1. Solve dV/dt = 8 − 0.4t, V(0) = 100.

2. A ball has velocity dh/dt = 15 − 9.8t and starts at h = 2 m. Find h(t).

6 Exponential growth: dy/dt = ky

Rate proportional to size.

A is the initial value y(0). k > 0 gives growth; k < 0 decay. The doubling time is ln 2/k.

WORKED EXAMPLE 3 Population (standard)

dP/dt = 0.025P, P(0) = 12 000. Find P(t), P(10) and the doubling time.

Solution

P = 12 000e0.025tA1

P(10) = 15 408A1

ln 2/0.025 = 27.7 yearsA1

Practice exercise 4

1. dN/dt = 0.3N, N(0) = 50. Find N(5).

2. A culture doubles every 4 hours. Find k in dN/dt = kN.

7 Exponential decay and half-life

k < 0.

WORKED EXAMPLE 4 Radioactive decay (standard)

dm/dt = −0.035m, m(0) = 80 g. Find m(20) and the half-life.

Solution

m = 80e−0.035t: m(20) = 39.7 gA1 A1

Half-life ln 2/0.035 = 19.8 yearsA1

Practice exercise 5

1. A drug's half-life is 6 hours. Find k and the fraction remaining after 15 hours.

8 Newton's law of cooling

Approach to room temperature.

WORKED EXAMPLE 5 Cooling (exam level)

Tea at 90 °C cools in a 20 °C room; after 5 minutes it is 70 °C. Find k and the time to reach 45 °C.

Solution

70 = 20 + 70e−5k: k = 0.0673M1 A1

45 = 20 + 70e−kt: t = 15.3 minM1 A1

Practice exercise 6

1. A body is found at 30 °C in a 10 °C room; an hour later it is 26 °C. Assuming 37 °C at death, estimate the time since death.

9 Separation of variables

Collect y on one side.

For dy/dx = f(x)g(y), rearrange to (1/g(y)) dy = f(x) dx and integrate both sides. Include one constant, then apply the initial condition.

WORKED EXAMPLE 6 Separable (standard)

Solve dy/dx = xy, y(0) = 2.

Solution

∫(1/y) dy = ∫x dx: ln y = x2/2 + cM1 A1

y = Aex²/2, A = 2A1

y = 2ex²/2A1

Practice exercise 7

1. Solve dy/dx = y/x, y(1) = 3 (x > 0).

2. Solve dy/dx = 2x/y, y(0) = 4 (y > 0).

10 Slope fields: construction

Short line segments.

A slope field draws, at many points (x, y), a short segment with gradient dy/dx given by the DE. To draw one by hand, compute the gradient at grid points; isoclines (curves with equal gradient) speed this up.

Figure 2. A slope field with the solution curve through (0, 1).

Practice exercise 8

1. Find the gradient of the slope field for dy/dx = x − y at (1, 3) and (2, 2).

2. Describe the slope field of dy/dx = y.

11 Slope fields: interpretation

Reading solution curves.

Solution curves follow the segments. From a slope field you can sketch the particular solution through a given point, identify equilibrium (horizontal) solutions, and describe long-term behaviour without solving the equation.

Figure 3. Solutions approach the equilibrium y = 4.

12 Euler's method

Step along tangents.

Starting at (x0, y0), move a small step h in x and use the gradient at the current point to estimate the change in y. Repeating gives an approximate solution.

WORKED EXAMPLE 7 Euler (standard)

Use Euler's method with h = 0.1 to estimate y(0.3) for dy/dx = x + y, y(0) = 1.

Solution

y1 = 1 + 0.1(1) = 1.1M1 A1

y2 = 1.1 + 0.1(1.2) = 1.22A1

y3 = 1.22 + 0.1(1.42) = 1.362A1

n	xn	yn	f(xn, yn)
0	0	1	1
1	0.1	1.1	1.2
2	0.2	1.22	1.42
3	0.3	1.362

Practice exercise 9

1. Use Euler with h = 0.5 to estimate y(1) for dy/dx = 2y, y(0) = 1, and compare with e2.

2. Estimate y(0.2) for dy/dx = x2 + y, y(0) = 2 with h = 0.1.

13 Euler's method with technology and step size

Smaller steps, better accuracy.

Use a spreadsheet or GDC sequence mode to do many steps. Halving h usually roughly halves the error. Euler underestimates when solutions are concave up and overestimates when concave down.

Step h	Estimate of y(1) for dy/dx = x + y, y(0) = 1	Error
0.1	3.187	0.249
0.05	3.306	0.130
0.01	3.409	0.027
exact	2e − 2 = 3.437

Practice exercise 10

1. Why does Euler's method underestimate y(1) here?

14 Equilibrium solutions and long-term behaviour

Where dy/dt = 0.

An equilibrium solution is a constant solution: dy/dt = 0. It is stable if nearby solutions move towards it and unstable if they move away. For cooling, T = Ts is stable; for dP/dt = kP(1 − P/K), P = K is stable and P = 0 unstable.

Practice exercise 11

1. Find and classify the equilibria of dy/dt = y(3 − y).

15 The logistic growth model

Growth with a limit.

K is the carrying capacity: the long-term population. Growth is fastest when P = K/2. A = (K − P0)/P0.

WORKED EXAMPLE 8 Logistic (exam level)

A fish population follows dP/dt = 0.4P(1 − P/2000), P(0) = 200. Find P(t), P(5) and when P = 1500.

Solution

A = 9: P = 2000/(1 + 9e−0.4t)A1

P(5) = 902A1

1 + 9e−0.4t = 4/3: t = 8.24M1 A1

Practice exercise 12

1. For the fish model, find the population at which growth is fastest and the maximum growth rate.

16 Choosing and evaluating a model

Does the DE fit reality?

Compare the model's predictions with data, consider whether assumptions are reasonable (constant rates, no migration, uniform temperature), and look at long-term behaviour. Exponential models fit early growth; logistic models are better when resources are limited.

17 Technology for differential equations

Slope fields and numerical solutions.

GDC tip

Many GDCs plot slope fields and numerical solutions (DIFF EQ or similar graph mode). Use sequence mode or a spreadsheet for Euler's method, recording the first few rows and the final value.

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 9 Mixed problem (exam level)

dV/dt = −k√V models water draining from a tank, V(0) = 400 L, V(10) = 225 L. Solve and find when the tank is empty.

Solution

∫V−1/2 dV = ∫−k dt: 2√V = −kt + 40M1 A1

t = 10: 30 = −10k + 40, k = 1A1

Empty when 2√V = 0: t = 40 minA1

Practice exercise 13

1. Solve dy/dx = 3y2, y(0) = 1, and state where the solution is undefined.

19 Coupled systems and phase portraits (HL)

Two interacting quantities.

Coupled systems model interacting populations (predator-prey, competing species) or quantities. Writing them as a matrix, the eigenvalues of [[a, b], [c, d]] determine the behaviour: both negative, a stable node (solutions approach the origin); both positive, unstable; opposite signs, a saddle; complex with negative real part, a stable spiral. The phase portrait plots y against x.

WORKED EXAMPLE 10 Eigenvalues (exam level)

For dx/dt = x + 2y, dy/dt = 2x + y, find the eigenvalues and classify the equilibrium.

Solution

det [[1 − λ, 2], [2, 1 − λ]] = 0: (1 − λ)2 = 4M1 A1

λ = 3, −1: saddle point (unstable)A1 R1

Practice exercise 14

1. Classify the equilibrium for dx/dt = −2x + y, dy/dt = x − 2y.

2. Use Euler with h = 0.1 for one step of dx/dt = y, dy/dt = −x from (1, 0).

20 Second-order differential equations (HL)

Acceleration models.

A second-order DE such as d2x/dt2 + 0.5 dx/dt + 4x = 0 (a damped spring) is written as a coupled system with v = dx/dt: dx/dt = v, dv/dt = −4x − 0.5v. Euler's method or technology then gives approximate solutions.

WORKED EXAMPLE 11 Damped oscillation (exam level)

Using h = 0.1 and x(0) = 1, v(0) = 0, perform two Euler steps for the spring above.

Solution

Step 1: x = 1, v = 0 + 0.1(−4) = −0.4M1 A1

Step 2: x = 1 + 0.1(−0.4) = 0.96, v = −0.4 + 0.1(−4 + 0.2) = −0.78A1

Practice exercise 15

1. Write d2y/dt2 = −9y as a coupled system.

21 Investigation: modelling the spread of a rumour

A Paper 3 style exploration.

In a school of 1200 students, 10 students start a rumour. The number N who have heard it satisfies dN/dt = 0.0008N(1200 − N), t in hours. 1. Identify the type of model and the carrying capacity. 2. Use Euler's method with h = 1 to estimate N after 5 hours. 3. Solve the logistic equation and compare with your Euler estimate. 4. Find when half the school has heard the rumour. 5. Suggest a limitation of the model.

Answers. 1. Logistic, K = 1200, k = 0.96. 2. Euler gives about 10 → 19.5 → 38.0 → 73.2 → 139 → 257. 3. N = 1200/(1 + 119e−0.96t): N(5) ≈ 606; Euler with h = 1 badly underestimates because the growth is so fast. 4. t = ln 119/0.96 = 4.98 hours. 5. Students do not mix uniformly; some never hear or ignore the rumour.

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 10 minutes)

The number of bacteria N satisfies dN/dt = 0.2N, N(0) = 500.

(a) Write down N(t). [2]

(b) Find N when t = 8. [1]

Markscheme

Part	Answer	Marks
(a)	N = 500e0.2t	A2
(b)	2477	A1
(c)	e0.2t = 10	M1
	t = 11.5	A1

Examiner's comments. Recognize the exponential solution directly.

Question 2 (Paper 2 style) (about 15 minutes)

Consider dy/dx = y − x with y(0) = 2.

(a) Use Euler's method with h = 0.2 to estimate y(1). [4]

(b) Show that y = x + 1 + ex is the solution. [3]

Markscheme

Part	Answer	Marks
(a)	table of 5 steps	M1
	y values 2.4, 2.84, 3.328, 3.874, 4.488	A2
	y(1) ≈ 4.49	A1
(b)	dy/dx = 1 + ex	A1
	y − x = 1 + ex	M1
	y(0) = 2	AG A1
(c)	exact 4.718	A1
	4.87%	A1

Examiner's comments. Present Euler steps in a table.

Question 3 (Paper 2 style) (about 10 minutes)

A deer population follows dP/dt = 0.3P(1 − P/800), P(0) = 100.

(a) State the carrying capacity. [1]

(b) Write down P(t). [2]

Markscheme

Part	Answer	Marks
(a)	800	A1
(b)	P = 800/(1 + 7e−0.3t)	A2
(c)	1 + 7e−0.3t = 4/3	M1
	t = 10.1	A1

Examiner's comments. A = (800 − 100)/100 = 7.

23 Common misconceptions and links

Students often think…	Actually…
dy/dt = ky gives y = ekt + c	y = Aekt
Euler's method is exact	it approximates
two constants after separating	one is enough
equilibria are always stable	check nearby behaviour

How 5.14 connects to the rest of the course

Topic	Connection
5.5 Integration	solving by integration
1.5 Exponential functions	growth and decay
5.3 Derivatives	rates of change
1.3 Geometric sequences	discrete growth

24 Check your understanding

Section A: multiple choice

1. The solution of dy/dt = 0.1y is

A. y = Ae0.1t B. y = 0.1t + c C. y = e0.1 + c D. y = 0.1y2

2. Euler's method uses

A. the exact solution B. the area C. the gradient at the current point D. the second derivative only

3. In dP/dt = kP(1 − P/K), K is

A. the growth rate B. the carrying capacity C. the initial value D. the half-life

4. An equilibrium has

A. y = 0 B. t = 0 C. dy/dt = 1 D. dy/dt = 0

5. Newton's cooling approaches

A. zero B. the surrounding temperature C. infinity D. the initial temperature

Section B: structured

6. Solve dy/dx = 4x3, y(1) = 2. [2]

7. Find the half-life if dm/dt = −0.08m. [2]

Answers

1. A 2. C 3. B 4. D 5. B

6. y = x4 + 1

7. 8.66

25 Summary and topic map

Idea	Remember
DE	relates a function to its derivative
Direct integration	dy/dx = f(x)
Exponential	dy/dt = ky ⇒ Aekt
Cooling	Ts + (T0 − Ts)e−kt
Separation	∫1/g(y) dy = ∫f(x) dx
Slope fields	segments with gradient f(x, y)
Euler	yn+1 = yn + hf(xn, yn)
Logistic	K/(1 + Ae−kt)

Before the exam, make sure you…

•include the constant once
•tabulate Euler steps
•identify equilibria
•interpret in context

Differential Equations on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 4. Topic map for Differential Equations.

26 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. dA/dt = −kA

2. dN/dt = k(5000 − N)

Practice exercise 2

1. dy/dx = 2x

2. dT/dt = −6e−0.1t = −0.1(60e−0.1t)

Practice exercise 3

1. V = 100 + 8t − 0.2t2

2. h = 2 + 15t − 4.9t2

Practice exercise 4

1. 224

2. k = ln 2/4 = 0.173

Practice exercise 5

1. k = 0.1155; e−1.733 = 0.177

Practice exercise 6

1. k = ln(20/16) = 0.223; 37 → 30: t = ln(27/20)/0.223 = 1.34 hours before discovery

Practice exercise 7

1. y = 3x

2. y2 = 2x2 + 16

Practice exercise 8

1. −2; 0

2. horizontal on y = 0; steeper as |y| increases; independent of x

Practice exercise 9

1. y = 1, 2, 4: estimate 4 vs 7.39

2. 2.2, then 2.2 + 0.1(0.01 + 2.2) = 2.421

Practice exercise 10

1. the solution curve is concave up, so tangents lie below it

Practice exercise 11

1. y = 0 unstable; y = 3 stable

Practice exercise 12

1. P = 1000; dP/dt = 0.4(1000)(0.5) = 200 fish per year

Practice exercise 13

1. −1/y = 3x − 1: y = 1/(1 − 3x), undefined at x = 1/3

Practice exercise 14

1. λ = −1, −3: stable node

2. (1, −0.1)

Practice exercise 15

1. dy/dt = v, dv/dt = −9y

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