DP Maths AA HL - Calculus - Differential Equations

DP Maths AA HL - Calculus - Differential Equations - Full notes

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DP Maths AA HL | Differential Equations

Differential Equations

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5.18 Differential Equations

DP MATHEMATICS AA · TOPIC 5: CALCULUS

5.18

Differential Equations

Full notes for HL · Syllabus AHL 5.18

Contents

1	What is a differential equation?
2	Slope fields
3	Euler's method
4	Separation of variables
5	Modelling growth, decay and cooling
6	Homogeneous equations: the substitution y = vx
7	The integrating factor method
8	Forming differential equations: more practice
9	Separable equations: more practice
10	Growth and decay models
11	Newton's law of cooling: more practice
12	Logistic growth
13	Homogeneous equations: more practice
14	Integrating factor: more practice
15	Euler's method: accuracy and step size
16	Slope fields: interpreting solutions
17	Mixed review: worked problems
18	Additional exam-style practice
19	Choosing a solution method
20	Modelling with differential equations: evaluation
21	Exam-style questions with markscheme
22	Common misconceptions and links
23	Check your understanding
24	Summary and topic map
25	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 Quick check.

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 and interpret first-order differential equations from context
sketch and interpret slope fields
use Euler's method to approximate solutions
solve separable differential equations, finding general and particular solutions
model growth, decay and cooling with differential equations
solve homogeneous equations dy/dx = f(y/x) using y = vx
solve linear equations dy/dx + P(x)y = Q(x) with an integrating factor

This booklet covers Additional Higher Level (AHL) content (AHL 5.18). It is studied by HL students only.

Guiding questions

How can a rule for the rate of change determine a whole function? How do we choose a method to solve a differential equation?

How this topic is examined

Differential equations are a regular HL long question: solve by separating variables or with an integrating factor (Paper 1), use Euler's method with a GDC or table (Paper 2), and interpret solutions in context.

Identify the type of equation first: separable, homogeneous, or linear. Include the constant of integration immediately after integrating.

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
Solve	find the values of the unknowns, using algebra or technology as allowed
Show that	obtain the given result; every step must be shown and the result must not be assumed
Hence / Hence or otherwise	use the previous result (or another valid method)
Sketch	a diagram showing the general shape and key features, labelled
Prove	a formal, rigorous argument from accepted facts
Justify / Explain	give valid reasons or evidence

How to use these notes

Each section teaches one part of the syllabus. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or proof of a key result, so you understand where formulas come from
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 on Paper 2
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); examples are graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet
Quick check	short questions to check the key ideas

1 What is a differential equation?

An equation involving a derivative.

Think about it

A hot drink cools faster when it is much hotter than the room. How could you write this as an equation about the rate of change of temperature?

A differential equation (DE) relates a function to its derivatives. A first-order DE involves only the first derivative, for example dy/dx = 2xy. A solution is a function satisfying the equation. The general solution contains an arbitrary constant c (a family of curves); a particular solution uses an initial condition, such as y(0) = 3, to fix c.

Forming a DE from context: 'the rate of change of P is proportional to P' gives dP/dt = kP; Newton's law of cooling, 'the rate of cooling is proportional to the temperature difference', gives dT/dt = −k(T − Troom).

WORKED EXAMPLE 1 Verifying a solution (basic)

Show that y = Aex² satisfies dy/dx = 2xy.

Solution

dy/dx = 2xAex²M1

= 2x × y, as requiredAG A1

Practice exercise 1

1. Write a DE: the rate of increase of a rumour's spread N is proportional to the number who have not heard it, out of 1000 people.

2. Verify that y = x2 + c/x satisfies x(dy/dx) + y = 3x2.

2 Slope fields

Visualizing solutions without solving.

A slope field draws a short line segment at many points (x, y), with gradient given by the DE. Solution curves follow the directions of the segments. A particular solution is the curve through the initial point.

Figure 1. Slope field for dy/dx = x + y, with the solution through (0, 1).

3 Euler's method

Step along the tangent.

Euler's method approximates a solution of dy/dx = f(x, y) from a starting point, taking small steps of size h along the tangent direction. Smaller steps give better approximations. If the solution curve is concave up, Euler's method underestimates; if concave down, it overestimates.

WORKED EXAMPLE 2 Euler's method (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(0 + 1) = 1.1M1 A1

y2 = 1.1 + 0.1(0.1 + 1.1) = 1.22A1

y3 = 1.22 + 0.1(0.2 + 1.22) = 1.362A1

The exact solution y = 2ex − x − 1 gives y(0.3) = 1.3997, so the estimate is too low (the curve is concave up).

GDC tip

Use a spreadsheet or the GDC sequence mode: xn+1 = xn + h and yn+1 = yn + h f(xn, yn). Record a table of values in your answer.

Practice exercise 2

1. Use Euler's method with h = 0.5 to estimate y(1) for dy/dx = y2 − x, y(0) = 1.

2. Explain how to improve the accuracy of Euler's method.

4 Separation of variables

Collect y on one side, x on the other.

A DE is separable if it can be written as dy/dx = f(x)g(y). Rearrange to (1/g(y)) dy = f(x) dx, integrate both sides, and include one constant. Then use the initial condition and, if possible, write y explicitly.

WORKED EXAMPLE 3 Separable DE (standard)

Solve dy/dx = 2xy, y(0) = 3.

Solution

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

y = Aex²A1

y(0) = 3: y = 3ex²A1

WORKED EXAMPLE 4 Another separable DE (exam level)

Solve dy/dx = (1 + y2)/x, x > 0, given y = 0 when x = 1.

Solution

∫1/(1 + y2) dy = ∫1/x dx: arctan y = ln x + cM1 A1

x = 1, y = 0: c = 0A1

y = tan(ln x)A1

Practice exercise 3

1. Solve dy/dx = y/x2, y(1) = e.

2. Solve dy/dx = ex−y, y(0) = 0.

3. Solve dy/dx = x√y, y(0) = 4.

5 Modelling growth, decay and cooling

Solve, then interpret.

WORKED EXAMPLE 5 Newton's law of cooling (exam level)

A cup of tea at 90 °C cools in a 20 °C room with dT/dt = −k(T − 20). After 5 minutes it is 70 °C. Find T(t) and the time to reach 40 °C.

Solution

∫1/(T − 20) dT = ∫−k dt: T = 20 + Ae−kt; A = 70M1 A1

50 = 70e−5k, k = (1/5)ln(7/5) = 0.0673A1

20 = 70e−kt: t = ln 3.5/k = 18.6 minutesM1 A1

WORKED EXAMPLE 6 Logistic-type model (exam level)

Solve dP/dt = 0.2P(1 − P/100), P(0) = 10, using 1/(P(1 − P/100)) = 1/P + 1/(100 − P).

Solution

∫[1/P + 1/(100 − P)] dP = ∫0.2 dt: ln(P/(100 − P)) = 0.2t + cM1 A1

t = 0: c = ln(1/9)A1

P = 100/(1 + 9e−0.2t), which approaches 100A1

Practice exercise 4

1. A population grows with dP/dt = 0.03P, P(0) = 2000. Find when it doubles.

2. Solve dv/dt = 10 − 0.5v, v(0) = 0, and find the terminal velocity.

6 Homogeneous equations: the substitution y = vx

When the gradient depends on y/x.

A DE of the form dy/dx = f(y/x) is homogeneous. Substitute y = vx, so dy/dx = v + x(dv/dx). The equation becomes v + x(dv/dx) = f(v), which is separable in v and x.

WORKED EXAMPLE 7 Homogeneous DE (exam level)

Solve dy/dx = (x2 + y2)/(xy), x > 0, y(1) = 2.

Solution

dy/dx = x/y + y/x; let y = vx: v + x(dv/dx) = 1/v + vM1 A1

x(dv/dx) = 1/v: ∫v dv = ∫1/x dx, v2/2 = ln x + cM1 A1

y2 = 2x2(ln x + 2): y = x√(2 ln x + 4)A1

Practice exercise 5

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

7 The integrating factor method

For linear first-order equations.

Why it works

Multiplying the equation by I(x) = e∫P dx makes the left side I(dy/dx) + IPy. Since I′ = IP, this is exactly the product rule derivative of Iy. So d/dx(Iy) = IQ, and integrating gives Iy = ∫IQ dx.

WORKED EXAMPLE 8 Integrating factor (standard)

Solve dy/dx + 2y = ex, y(0) = 1.

Solution

I = e2x; d/dx(e2xy) = e3xM1 A1

e2xy = e3x/3 + cA1

y(0) = 1: c = 2/3, y = ex/3 + (2/3)e−2xA1

WORKED EXAMPLE 9 Rearranging first (exam level)

Solve x(dy/dx) − y = x3, x > 0, y(1) = 2.

Solution

dy/dx − y/x = x2; I = e−ln x = 1/xM1 A1

d/dx(y/x) = x: y/x = x2/2 + cA1

c = 3/2: y = x3/2 + 3x/2A1

Common mistake

Forgetting to divide by the coefficient of dy/dx before finding the integrating factor.

Practice exercise 6

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

2. Solve dy/dx + y tan x = cos x, y(0) = 2, for −π/2 < x < π/2.

8 Forming differential equations: more practice

Words to equations.

'The rate of change of y is proportional to y' gives dy/dt = ky. 'Inversely proportional to' gives dy/dt = k/y. 'Proportional to the difference between T and 20' gives dT/dt = k(T − 20). Decide the sign of k from whether the quantity increases or decreases.

WORKED EXAMPLE 10 Forming (standard)

The rate at which a liquid evaporates is proportional to the square root of the volume remaining. Write a DE.

Solution

dV/dt = −k√V, k > 0A1

Practice exercise 7

1. A rumour spreads at a rate proportional to the product of the number who know it, N, and the number who do not, in a population of 2000.

2. The rate of increase of an investment is 4% of its value per year.

9 Separable equations: more practice

Separate, integrate, solve for y.

Write the equation as g(y) dy = f(x) dx, integrate both sides, add one constant, use the initial condition, and make y the subject if possible.

WORKED EXAMPLE 11 Separable (exam level)

Solve dy/dx = (1 + y2)x, y(0) = 1.

Solution

∫1/(1 + y2) dy = ∫x dx: arctan y = x2/2 + cM1 A1

c = π/4A1

y = tan(x2/2 + π/4)A1

WORKED EXAMPLE 12 Exponential form (standard)

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

Solution

ln|y| = sin x + cM1 A1

y = 3esin xA1

Practice exercise 8

1. Solve dy/dx = eyx, y(0) = 0.

2. Solve dy/dx = (y + 1)/(x + 1), y(0) = 1.

10 Growth and decay models

Exponential solutions.

dN/dt = kN gives N = N0ekt. Find k from data at two times, then use the model to predict, and comment on its validity.

WORKED EXAMPLE 13 Bacteria (standard)

A culture grows from 500 to 1400 in 3 hours, with dN/dt = kN. Find N after 8 hours.

Solution

e3k = 2.8: k = 0.343M1 A1

N(8) = 500e8k = 7787A1

Practice exercise 9

1. Carbon-14 has a half-life of 5730 years. Find k in dN/dt = −kN.

2. A population of 8000 decreases by 3% per year continuously. Find when it halves.

11 Newton's law of cooling: more practice

Two data points.

T = Ts + (T0 − Ts)e−kt. Use one data point to find k, then predict temperatures or times.

WORKED EXAMPLE 14 Cooling (exam level)

A cake at 180 °C cools in a 20 °C kitchen to 120 °C after 10 min. When is it 40 °C?

Solution

100 = 160e−10k: k = 0.0470M1 A1

20 = 160e−kt: t = 44.2 minM1 A1

Practice exercise 10

1. A drink at 5 °C warms in a 25 °C room to 12 °C in 15 min. Find its temperature after 40 min.

12 Logistic growth

Growth limited by a carrying capacity.

dP/dt = kP(1 − P/K) has solution P = K/(1 + Ae−kt), with A = (K − P0)/P0. The growth rate is greatest when P = K/2, and P approaches K.

Figure 2. Logistic growth towards K = 1000.

WORKED EXAMPLE 15 Logistic (exam level)

P = 1000/(1 + 19e−0.5t). Find P(0), the time when P = 500 and the maximum growth rate.

Solution

P(0) = 50A1

19e−0.5t = 1: t = 5.89M1 A1

At P = 500: dP/dt = 0.5(500)(0.5) = 125 per unit timeA1

Practice exercise 11

1. Solve dP/dt = 0.2P(1 − P/400), P(0) = 40.

13 Homogeneous equations: more practice

y = vx.

For dy/dx = f(y/x), substitute y = vx so that dy/dx = v + x dv/dx. The equation becomes separable in v and x.

WORKED EXAMPLE 16 Homogeneous (exam level)

Solve dy/dx = (y2 + x2)/(2xy), y(1) = 2 (x > 0).

Solution

v + xv′ = (v2 + 1)/(2v): xv′ = (1 − v2)/(2v)M1 A1

∫2v/(1 − v2) dv = ∫1/x dx: −ln|1 − v2| = ln x + cM1

|1 − v2| = 3/x: with y(1) = 2, v2 = 1 + 3/x: y2 = x2 + 3xA1

Practice exercise 12

1. Solve dy/dx = 1 + y/x, y(1) = 2.

14 Integrating factor: more practice

Linear first-order equations.

Rewrite as dy/dx + P(x)y = Q(x), find I = e∫P dx, then (Iy)′ = IQ, integrate, and apply the initial condition.

WORKED EXAMPLE 17 Integrating factor (exam level)

Solve dy/dx − 2y/x = x3, y(1) = 2 (x > 0).

Solution

I = e−2 ln x = x−2A1

(y/x2)′ = x: y/x2 = x2/2 + cM1 A1

c = 3/2: y = x4/2 + 3x2/2A1

Practice exercise 13

1. Solve dy/dx + 3y = e−x, y(0) = 1.

2. Solve x dy/dx + y = sin x, y(π) = 0.

15 Euler's method: accuracy and step size

Smaller h, smaller error.

Euler's method approximates the solution by stepping along tangents. The error is roughly proportional to h: halving h roughly halves the error. Concave-up solutions are underestimated; concave-down overestimated.

h	Estimate of y(1) for y′ = y, y(0) = 1	Error
0.5	2.25	0.468
0.25	2.441	0.277
0.1	2.594	0.125
0.01	2.705	0.0135
exact	e = 2.718

WORKED EXAMPLE 18 Euler (standard)

Use Euler with h = 0.5 to estimate y(1) for y′ = y, y(0) = 1.

Solution

y1 = 1.5, y2 = 2.25A1

Underestimate since the solution ex is concave upR1

Practice exercise 14

1. Use Euler with h = 0.2 to estimate y(0.6) for y′ = x − y, y(0) = 1.

16 Slope fields: interpreting solutions

Without solving.

From a slope field, sketch solution curves, identify equilibrium solutions (horizontal lines where dy/dx = 0 for all x) and their stability, and describe the long-term behaviour.

WORKED EXAMPLE 19 Equilibria (standard)

For dy/dx = y(2 − y), identify the equilibria and their stability.

Solution

y = 0 (unstable) and y = 2 (stable)A1 A1

Practice exercise 15

1. Describe the solution of dy/dx = y(2 − y) with y(0) = 0.5 as x → ∞.

17 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 20 Mixed problem (exam level)

A tank contains 100 L of water with 10 kg of salt. Pure water flows in at 5 L/min and the mixture flows out at the same rate. (a) Show dS/dt = −S/20. (b) Find S(t) and when S = 1 kg.

Solution

(a) Out: concentration S/100 × 5 L/minM1 AG

(b) S = 10e−t/20A1

10e−t/20 = 1: t = 46.1 minM1 A1

Practice exercise 16

1. If the inflow contains 0.2 kg/L of salt instead, write and solve the DE.

18 Additional exam-style practice

Paper 1 and Paper 2.

WORKED EXAMPLE 21 Paper 1 question (exam level)

Solve (x2 + 1) dy/dx = 2xy, y(0) = 3.

Solution

∫1/y dy = ∫2x/(x2 + 1) dxM1

ln y = ln(x2 + 1) + cA1

y = 3(x2 + 1)A1

Practice exercise 17

1. Solve dy/dx = 2x(y − 1), y(0) = 3.

19 Choosing a solution method

Identify the type.

Form	Method
dy/dx = f(x)	integrate directly
dy/dx = f(x)g(y)	separate variables
dy/dx = f(y/x)	substitute y = vx
dy/dx + P(x)y = Q(x)	integrating factor
cannot be solved exactly	Euler's method or technology

WORKED EXAMPLE 22 Choosing (standard)

Choose a method for dy/dx = (x + y)/x and for dy/dx = xy2.

Solution

First: homogeneous (or linear); second: separableR1 R1

20 Modelling with differential equations: evaluation

Assumptions and limitations.

Every DE model makes assumptions: constant rates, a well-mixed tank, unlimited resources (exponential) or a fixed carrying capacity (logistic). Compare predictions with data and discuss when the model breaks down.

WORKED EXAMPLE 23 Evaluating (standard)

Why is the exponential model dN/dt = kN unrealistic for a population over a long time?

Solution

It predicts unlimited growth; resources are limitedR1

21 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 15 minutes)

Consider dy/dx = y cos x, with y = 2 when x = 0.

(a) Solve the differential equation. [5]

(b) Use Euler's method with h = 0.5 to estimate y(1). [3]

Markscheme

Part	Answer	Marks
(a)	∫1/y dy = ∫cos x dx	M1
	ln y = sin x + c	A1 A1
	c = ln 2	A1
	y = 2esin x	A1
(b)	y1 = 2 + 0.5(2 cos 0) = 3	M1 A1
	y2 = 3 + 0.5(3 cos 0.5) = 4.32	A1
(c)	exact 2esin 1 = 4.64; the estimate is too low because h is large	R1

Examiner's comments. Comparing with the exact value is the expected comment.

Question 2 (Paper 1 style) (about 10 minutes)

Solve (x2 + 1)(dy/dx) + 2xy = 4x, given y(0) = 1. [6]

Markscheme

Part	Answer	Marks
	d/dx((x2 + 1)y) = 4x (exact derivative, or I = x2 + 1)	M1 A1
	(x2 + 1)y = 2x2 + c	A1
	c = 1	A1
	y = (2x2 + 1)/(x2 + 1)	A1 A1

Examiner's comments. Recognizing the left side as a product derivative is the key step.

22 Common misconceptions and links

Students often think…	Actually…
add + c to both sides separately	one constant is enough
Euler's method gives the exact answer	it approximates
use the integrating factor without dividing first	make the coefficient of dy/dx equal 1
ln\|y\| = x2 + c gives y = ex² + c	y = Aex²

How 5.18 connects to the rest of the course

Topic	Connection
5.16 Further integration	integration techniques
2.8 Exponential models	growth and decay
1.11 Partial fractions	logistic equations
5.19 Maclaurin series	series solutions

23 Check your understanding

Section A: multiple choice

1. dy/dx = 3y has general solution

A. y = Ae3x B. y = 3ex C. y = e3x + c D. y = 3x + c

2. The integrating factor for dy/dx + 4y = x is

A. ex B. 4x C. e4x D. e4

3. Euler's method uses

A. the exact solution B. the tangent gradient at each step C. the second derivative D. integration

4. A homogeneous DE is solved by

A. separating directly B. an integrating factor C. Euler's method only D. y = vx

Section B: structured

5. Solve dy/dx = 3x2y2, y(0) = 1. [4]

6. Solve dy/dx − 2y = 4, y(0) = 1. [4]

Answers

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

5. −1/y = x3 − 1: y = 1/(1 − x3).

6. I = e−2x: y = 3e2x − 2.

24 Summary and topic map

Idea	Remember
DE	equation with derivatives; general and particular solutions
Slope field	segments showing gradients
Euler	yn+1 = yn + hf(xn, yn)
Separable	∫1/g(y) dy = ∫f(x) dx
Homogeneous	y = vx
Linear	integrating factor e∫P dx
Models	growth, decay, cooling, logistic

Before the exam, make sure you…

•identify the type first
•include the constant immediately
•divide to make the dy/dx coefficient 1
•interpret solutions 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 3. Topic map for Differential Equations.

25 Answers to practice exercises

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

Practice exercise 1

1. dN/dt = k(1000 − N)

2. x(2x − c/x2) + x2 + c/x = 3x2

Practice exercise 2

1. y1 = 1.5; y2 = 1.5 + 0.5(2.25 − 0.5) = 2.375

2. use a smaller step size h

Practice exercise 3

1. ln y = −1/x + 2: y = e2 − 1/x

2. ey = ex: y = x

3. 2√y = x2/2 + 4: y = (x2/4 + 2)2

Practice exercise 4

1. ln 2/0.03 = 23.1

2. v = 20(1 − e−0.5t); 20

Practice exercise 5

1. v + x dv/dx = 1 + v: v = ln x: y = x ln x

Practice exercise 6

1. xy = x3/3 + 2/3: y = x2/3 + 2/(3x)

2. I = sec x: y sec x = x + 2: y = (x + 2)cos x

Practice exercise 7

1. dN/dt = kN(2000 − N)

2. dA/dt = 0.04A

Practice exercise 8

1. −e−y = x2/2 − 1: y = −ln(1 − x2/2)

2. y + 1 = 2(x + 1): y = 2x + 1

Practice exercise 9

1. 1.21 × 10−4

2. ln 2/0.03 = 23.1 years

Practice exercise 10

1. k = ln(20/13)/15 = 0.0287; T = 25 − 20e−40k = 18.6 °C

Practice exercise 11

1. P = 400/(1 + 9e−0.2t)

Practice exercise 12

1. v = ln x + 2: y = x ln x + 2x

Practice exercise 13

1. y = ½e−x + ½e−3x

2. xy = −cos x + c: c = −1: y = −(1 + cos x)/x

Practice exercise 14

1. 0.8, 0.68, 0.624

Practice exercise 15

1. it increases towards 2

Practice exercise 16

1. dS/dt = 1 − S/20: S = 20 − 10e−t/20

Practice exercise 17

1. y = 1 + 2ex²

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