Differential Equations - IB Math AA Question Bank

Subjects

Features

Review

HOT

Tutoring

Differential Equations

Practice exam-style IB Math AA questions for Differential Equations, aligned with the syllabus and grouped by topic.

Verified by Karim

Paper

Difficulty

Status

Level

This content can't be printed from the browser. Use Export PDF with Pro for an offline copy.

Question 1

HL • Paper 1

Easy

Non Calculator

HL • Paper 1

Easy

Non Calculator

A solution of the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}x}=x-y$ satisfies $y(0)=2$ . Euler's method is used with step length $h=\dfrac12$ .

Write down the recurrence formulae for $x_{n+1}$ and $y_{n+1}$ .

[2]

Hence find the Euler approximation to $y(1)$ .

[3]

Question 2

HL • Paper 1

Easy

Non Calculator

HL • Paper 1

Easy

Non Calculator

Euler's method is used to approximate the solution of $\dfrac{\mathrm{d}y}{\mathrm{d}x}=x+y$ , with $y(0)=1$ , using step length $h=\dfrac12$ .

Write down the recurrence formulae used in Euler's method.

[2]

Use the recurrence formulae to find the Euler approximation to $y(1)$ .

[2]

Question 3

HL • Paper 2

Easy

Calculator Permitted

HL • Paper 2

Easy

Calculator Permitted

Consider the differential equation

\frac{dy}{dx}=x-y^2

with initial condition $y(0)=1$ .

Use Euler's method with step length $h=0.2$ to find an approximation for $y(0.6)$ . Give your answer correct to three significant figures.

[5]

Question 4

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $y$ satisfies $\dfrac{\mathrm{d}y}{\mathrm{d}x}=2xy^2$ and $y(0)=1$ .

Find the general solution in implicit form.

[3]

Hence find the particular solution in the form $y=f(x)$ .

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 5

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

For $x>0$ , consider the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}x}+\dfrac{2}{x}y=x^2$ , with $y(1)=0$ .

Find an integrating factor for this differential equation.

[2]

Hence solve the differential equation.

[4]

Question 6

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $y$ satisfies $\dfrac{\mathrm{d}y}{\mathrm{d}x}-y=e^x$ and $y(0)=2$ .

Find an integrating factor and write the left hand side as the derivative of a product.

[2]

Hence find $y$ in terms of $x$ .

[3]

Question 7

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

For $x>0$ , a function $y$ satisfies $\dfrac{\mathrm{d}y}{\mathrm{d}x}=1+\dfrac{y}{x}$ , with $y(1)=0$ .

Use the substitution $y=vx$ to find $v$ in terms of $x$ .

[3]

Hence find the particular solution for $y$ .

[2]

Question 8

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $y$ satisfies $\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{1+x}{1+y}$ and $y(0)=1$ .

Find an implicit expression relating $x$ and $y$ .

[3]

Hence write the solution explicitly in the form $y=f(x)$ , choosing the branch consistent with the initial condition.

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 9

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A curve satisfies the differential equation

\frac{dy}{dx}=xe^{-y}

and passes through the point $(0,\ln 2)$ .

Find $y$ in terms of $x$ , and hence find $y(2)$ .

[5]

Question 10

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

For $x>0$ , a curve satisfies

\frac{dy}{dx}=1+\frac{y}{x}

and passes through $(1,2)$ .

Using the substitution $y=vx$ , solve the differential equation and find $y(3)$ .

[5]

Question 11

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The temperature $T$ , in degrees Celsius, of a heated object is modelled by

\frac{dT}{dt}=-0.08(T-18)

where $t$ is measured in minutes. Initially $T=90$ .

Determine the time taken for the object to cool to $30^\circ\text{C}$ .

[5]

Question 12

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A radioactive substance has mass $M$ grams after $t$ years. Its decay is modelled by

\frac{dM}{dt}=-kM, \qquad k>0

After $200$ years, $65\%$ of the original mass remains.

Find $k$ and hence find the half-life of the substance.

[5]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 13

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A population $n$ is modelled by the logistic differential equation $\dfrac{\mathrm{d}n}{\mathrm{d}t}=\dfrac1{12}n(6-n)$ , where $0<n<6$ . Initially $n=2$ when $t=0$ .

Show that $\ln\left(\dfrac{n}{6-n}\right)=\dfrac{t}{2}+C$ .

[4]

Determine the value of $t$ when $n=3$ .

[2]

Question 14

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

For $x>0$ , a function $y$ satisfies $\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{y}{x}+\dfrac{x}{y}$ and $y(1)=2$ .

Use the substitution $y=vx$ to obtain a separable differential equation in $v$ and $x$ .

[3]

Hence solve the differential equation, giving $y$ in terms of $x$ .

[3]

Question 15

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The temperature $T$ , in degrees Celsius, of a cup of tea after $t$ minutes is modelled by $\dfrac{\mathrm{d}T}{\mathrm{d}t}=-k(T-20)$ , where $k>0$ . Initially $T=80$ , and after one minute $T=50$ .

Solve the differential equation in terms of $k$ .

[2]

Find the exact value of $k$ .

[2]

Determine the time at which $T=35$ .

[2]

Question 16

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

For $x>0$ , the function $y$ satisfies

\frac{dy}{dx}=\left(\frac{y}{x}\right)^2+\frac{y}{x}

with $y(1)=1$ .

Use the substitution $y=vx$ to find $y$ in terms of $x$ . State the vertical asymptote of the solution curve.

[6]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 17

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The function $y$ satisfies the differential equation

\frac{dy}{dx}+2y=e^{-x}

with $y(0)=3$ .

Solve the differential equation, and hence find the value of $x$ for which $y=0.2$ .

[6]

Question 18

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

For $x>0$ , a function $y$ satisfies

\frac{dy}{dx}+\frac{2}{x}y=\sin x

and $y(1)=0$ .

Find an expression for $y$ involving a definite integral, and hence find $y(3)$ .

[6]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

For $x>0$ , a curve satisfies

\frac{dy}{dx}=\frac{x^2+y^2}{xy}

and passes through $(1,1)$ .

Use the substitution $y=vx$ to solve the differential equation, and hence find $y(4)$ .

[6]

Question 20

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A quantity $S$ satisfies the differential equation

\frac{dS}{dt}+0.1S=5+2e^{-0.2t}

where $t\ge 0$ , and $S(0)=20$ .

Determine $S$ in terms of $t$ , and find the limiting value of $S$ as $t\to\infty$ .

[6]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 21

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A curve satisfies the differential equation

\frac{dy}{dx}=x-y

with initial condition $y(0)=1$ .

Euler's method with step length $h=\frac12$ is used to approximate $y(1)$ .

I.

Write down the recurrence formulae for $x_{n+1}$ and $y_{n+1}$ .

[2]

II.

Hence find the Euler approximation to $y(1)$ .

[3]

Solve the differential equation exactly, giving $y$ in terms of $x$ .

[5]

Determine whether the Euler approximation found in part (a) is greater than or less than the exact value of $y(1)$ . Justify your answer.

[1]

Question 22

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A function $y$ satisfies

\frac{dy}{dx}=x(1+y^2), \qquad y(0)=0

Solve the differential equation, giving $y$ explicitly in terms of $x$ .

[5]

I.

Find the least positive value of $x$ for which $y=1$ .

[2]

II.

Find the first positive vertical asymptote of the solution curve.

[2]

III.

State the largest interval containing $0$ on which this solution is continuous.

[1]

Question 23

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

For $x>0$ , the function $y$ satisfies $x\dfrac{\mathrm{d}y}{\mathrm{d}x}+(x+1)y=x^2e^{-x}$ , with $y(1)=0$ .

Write the differential equation in the form $y'+P(x)y=Q(x)$ and find an integrating factor.

[3]

Hence solve the differential equation.

[4]

Question 24

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A quantity $n$ is modelled by $\dfrac{\mathrm{d}n}{\mathrm{d}t}=\dfrac1{100}n(20-n)$ , where $0<n<20$ . Initially $n=5$ when $t=0$ .

Show that the general solution may be written as $\ln\left(\dfrac{n}{20-n}\right)=\dfrac{t}{5}+C$ .

[3]

Find $n$ explicitly in terms of $t$ .

[3]

Determine the value of $t$ when $n=10$ .

[1]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 25

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A curve satisfies

\frac{dy}{dx}=\frac{x^2+1}{y+3}, \qquad y(0)=1

Solve the differential equation, giving $y$ explicitly in terms of $x$ .

[4]

Use Euler's method with step length $h=1$ to approximate $y(3)$ .

[2]

I.

Find the exact value of $y(3)$ .

[2]

II.

Determine the value of $x$ for which $y=5$ .

[2]

Question 26

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A population $P$ is modelled by the logistic differential equation

\frac{dP}{dt}=kP(500-P), \qquad k>0

where $t$ is measured in days. Initially $P=50$ , and after $10$ days $P=150$ .

Determine the value of $k$ and hence find the time at which $P=400$ .

[7]

Question 27

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The differential equation

\frac{dy}{dx}=y+x

has initial condition $y(0)=1$ .

Use Euler's method with step length $h=0.5$ to approximate $y(1)$ . Then solve the differential equation exactly and calculate the percentage error in the Euler approximation.

[7]

Question 28

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A population $N$ , where $0<N<10$ , is modelled by

\frac{dN}{dt}=\frac1{20}N(10-N), \qquad N(0)=2

I.

Separate variables and integrate to obtain an implicit relation between $N$ and $t$ .

[3]

II.

Hence find $N$ explicitly in terms of $t$ .

[4]

I.

The rate of growth is greatest when $N=a$ . Find $a$ .

[2]

II.

Find the exact time at which the rate of growth is greatest.

[2]

Determine $\lim_{t\to\infty}N(t)$ and interpret this value in the context of the model.

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 29

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $x>0$ , a function $y$ satisfies

\frac{dy}{dx}-\frac{2}{x}y=x^2\cos x, \qquad y\left(\frac{\pi}{2}\right)=0

I.

Find an integrating factor for the differential equation.

[2]

II.

Hence solve the differential equation.

[4]

Consider the solution curve on the interval $0<x<\pi$ .

I.

Show that $x=\frac{\pi}{2}$ is a stationary point of the curve.

[2]

II.

Determine whether this stationary point is a local maximum or a local minimum. Justify your answer.

[2]

Question 30

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A function $y$ satisfies

\frac{dy}{dx}+y=e^xy^2, \qquad y(0)=1

Let $z=\frac1y$ .

I.

Show that $z$ satisfies the linear differential equation $z'-z=-e^x$ .

[3]

II.

Solve the differential equation for $y$ in terms of $x$ .

[4]

I.

Find the exact value of $y(\ln2)$ .

[1]

II.

State the vertical asymptote of the solution and justify your answer.

[2]

Question 31

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The function $y$ satisfies

\frac{dy}{dx}=y(2-x), \qquad y(0)=1

Use Euler's method with step length $h=\frac12$ to approximate $y(1)$ .

[4]

I.

Solve the differential equation exactly.

[3]

II.

Find the exact value of $y(1)$ .

[1]

Without using a calculator, determine whether the Euler approximation is greater than or less than the exact value.

[3]

Question 32

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A quantity $P$ satisfies

\frac{dP}{dt}=\frac1{12}(P-3)(9-P), \qquad 3<P<9

and $P=5$ when $t=0$ .

I.

Find an implicit relation between $P$ and $t$ .

[4]

II.

Hence find $P$ explicitly in terms of $t$ .

[3]

I.

Find the exact time at which $P=7$ .

[2]

II.

State the two equilibrium values and determine which one is approached by $P(t)$ as $t\to\infty$ .

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 33

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $x>0$ , a curve satisfies

\frac{dy}{dx}=\frac{y}{x}+\left(\frac{y}{x}\right)^3, \qquad y(1)=1

Use the substitution $y=vx$ .

I.

Find a separable differential equation in $v$ and $x$ .

[3]

II.

Hence find $y$ in terms of $x$ .

[5]

I.

State the largest interval containing $1$ on which this solution is defined.

[2]

II.

Find the gradient of the curve at $(1,1)$ .

[1]

Question 34

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A function $y$ satisfies

\frac{dy}{dx}=2x(1-y), \qquad y(0)=0

Use Euler's method with step length $h=\frac12$ to approximate $y(1)$ .

[4]

I.

Solve the differential equation exactly.

[4]

II.

Find the exact value of $y(1)$ .

[1]

Determine whether the Euler approximation is greater than or less than the exact value. Justify your answer without using a calculator.

[2]

Question 35

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A colony of bacteria is modelled by the logistic differential equation

\frac{dP}{dt}=0.4P(5-P), \qquad 0<P<5

where $P$ is measured in thousands and $t$ is measured in hours. Initially $P=1.2$ when $t=0$ .

Use Euler's method with step length $h=1$ to find an approximation for $P(4)$ .

[3]

I.

Solve the differential equation to show that

P=\frac{5}{1+\frac{19}{6}e^{-2t}}

[4]

II.

Hence find the exact-model value of $P(4)$ .

[2]

Using your solution in part (b), determine the time at which the population first reaches $4.5$ thousand.

[3]

Question 36

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The temperature $T$ , in degrees Celsius, of an industrial component after $t$ minutes is modelled by

\frac{dT}{dt}+0.12T=2.4+4e^{-0.05t}, \qquad T(0)=70

Part (a)

I.

Write down the Euler recurrence formulae for step length $h=5$ .

[2]

II.

Use these formulae to approximate $T(15)$ .

[2]

Solve the differential equation exactly.

[4]

Part (c)

I.

Use your exact solution to calculate $T(15)$ .

[2]

II.

Determine the time at which $T=30$ .

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 37

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A population $N$ is modelled by

\frac{dN}{dt}=kN(800-N), \qquad k>0

where $t$ is measured in days. Initially $N=40$ , and after $6$ days $N=160$ .

I.

Solve the differential equation and determine the value of $k$ .

[4]

II.

Write $N$ explicitly as a function of $t$ .

[2]

Determine the time at which $N=600$ .

[3]

Using the unrounded value of $k$ found in part (a), use Euler's method with step length $h=2$ to approximate $N(6)$ , and comment on the approximation.

[2]

Question 38

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $x>0$ , the function $y$ satisfies

x\frac{dy}{dx}+(1-2x)y=x^2, \qquad y(1)=3

Write the differential equation in standard linear form and find an integrating factor.

[4]

Hence solve the differential equation.

[3]

I.

Find $y(1.5)$ .

[2]

II.

Determine the values of $x$ for which $y=10$ .

[2]

Question 39

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The amount $M$ , in milligrams, of a medicine in the bloodstream after $t$ hours is modelled by

\frac{dM}{dt}+0.4M=20e^{-0.1t}, \qquad M(0)=0

Solve the differential equation.

[4]

I.

Determine the time at which $M$ is a maximum.

[2]

II.

Find the maximum amount of medicine in the bloodstream.

[3]

Use Euler's method with step length $h=1$ to approximate $M(3)$ , and compare this with the exact value.

[2]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $x\ge 0$ , a function $y$ satisfies

\frac{dy}{dx}+\frac{2x}{1+x^2}y=\frac{\cos x}{1+x^2}, \qquad y(0)=4

Find the exact solution for $y$ in terms of $x$ .

[3]

I.

Use Euler's method with step length $h=0.25$ to approximate $y(1)$ .

[3]

II.

Find the exact value of $y(1)$ and calculate the percentage error in the Euler approximation.

[2]

I.

Determine the $x$ -coordinate of the maximum point of the exact solution for $0\le x\le 4$ .

[2]

II.

Find the maximum value of $y$ on this interval.

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 41

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A model for the height $y$ metres of a fast-growing plant after $x$ weeks satisfies

\frac{dy}{dx}=y(2-x), \qquad y(0)=1, \qquad 0\le x\le 1

The diagram shows four representative slope segments for this differential equation on the interval $0\le x\le 1$ .

I.

Write down the recurrence formulae for Euler's method with step length $h=0.25$ .

[2]

II.

Use the recurrence formulae to find the Euler approximation to $y(0.5)$ .

[2]

Solve the differential equation exactly, giving $y$ in terms of $x$ .

[4]

I.

Continue Euler's method to $x=1$ and find the percentage error in the approximation. If you did not obtain an exact solution in part (b), use $y=e^{2x-\frac{x^2}{2}}$ .

[2]

II.

Justify, using calculus, why Euler's method with this step length gives an underestimate on $0\le x\le 1$ .

[2]

Question 42

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The number $N$ of cells, in thousands, in a culture is modelled by the logistic differential equation

\frac{dN}{dt}=kN(1000-N), \qquad 0<N<1000

where $t$ is measured in days. Initially $N=100$ , and after $5$ days $N=250$ .

I.

Show that the general solution can be written in the form

\ln\left(\frac{N}{1000-N}\right)=1000kt+C

[3]

II.

Determine the exact value of $k$ .

[2]

Find $N$ explicitly as a function of $t$ .

[3]

I.

Determine the time when $N=750$ .

[2]

II.

Find the maximum growth rate predicted by the model, and state when it occurs.

[2]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A tank initially contains $50$ litres of solution with $5$ kg of dissolved salt. Brine enters at $3$ litres per minute with salt concentration $0.4$ kg per litre. The well-mixed solution leaves at $2$ litres per minute.

Let $S$ be the mass of salt, in kg, in the tank after $t$ minutes.

I.

Explain why $S$ satisfies

\frac{dS}{dt}=1.2-\frac{2S}{50+t}

[2]

II.

Write the differential equation in linear form and find an integrating factor.

[2]

Solve for $S$ in terms of $t$ .

[4]

I.

Find the time when the concentration in the tank first reaches $0.35$ kg per litre.

[2]

II.

Interpret the limiting concentration predicted by the model.

[2]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A metal probe has temperature $T$ degrees Celsius at time $t$ minutes. The surrounding air temperature is rising linearly and is given by

A(t)=18+0.5t

The probe is modelled by Newton's law of cooling,

\frac{dT}{dt}=-0.1(T-A(t)), \qquad T(0)=40

I.

Write the differential equation in the form $T'+P(t)T=Q(t)$ .

[1]

II.

Find the integrating factor and hence solve for $T$ in terms of $t$ .

[3]

Find the time when the probe and the surrounding air have the same temperature.

[3]

I.

Show that, after the time found in part (b), the probe remains cooler than the surrounding air.

[2]

II.

Find the limiting value of $T-A$ as $t\to\infty$ and interpret it in context.

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $x>0$ , a curve satisfies

\frac{dy}{dx}=2\frac{y}{x}-\frac{x}{y}, \qquad y(1)=2

The curve lies above the $x$ -axis.

I.

Use the substitution $y=vx$ to form a separable differential equation.

[2]

II.

Hence show that $v^2-1=Cx^2$ , where $C$ is a constant.

[2]

Find $y$ explicitly in terms of $x$ .

[3]

Find the value of $x$ at which the gradient of the curve is $3$ .

[3]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A family of solution curves is defined by

\frac{dy}{dx}=x(1-y^2), \qquad y(0)=a

where $0<a<1$ .

Student-facing visual

None

I.

Separate the variables and integrate to obtain an implicit solution.

[3]

II.

Hence write $y$ explicitly in terms of $x$ and $a$ .

[2]

For $a=0.2$ , find the positive value of $x$ when $y=0.8$ .

[3]

Justify why no solution with $0<a<1$ crosses the line $y=1$ .

[3]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The differential equation

\frac{dy}{dx}=x^2-y, \qquad y(0)=1

is solved exactly and also approximated using Euler's method.

Step length $h$	Steps to $x=1$	Step indices	Abscissas $x_n$	Euler update
$0.5$	2	$n=0,1$	$0,\,h,\,2h=1$	$x_{n+1}=x_n+h,\quad y_{n+1}=y_n+h(x_n^2-y_n)$
$0.25$	4	$n=0,1,2,3$	$0,\,h,\,2h,\,3h,\,4h=1$	$x_{n+1}=x_n+h,\quad y_{n+1}=y_n+h(x_n^2-y_n)$

I.

Use Euler's method with $h=0.5$ to approximate $y(1)$ .

[2]

II.

Use Euler's method with $h=0.25$ to approximate $y(1)$ .

[2]

Solve the differential equation exactly and find the exact value of $y(1)$ .

[5]

Let $E_{0.5}$ and $E_{0.25}$ denote the absolute errors in the Euler approximations with step lengths $0.5$ and $0.25$ , respectively. Find the ratio $\frac{E_{0.5}}{E_{0.25}}$ . Comment on what this suggests about halving the step length.

[3]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A falling object has velocity $v$ metres per second after $t$ seconds. A simplified air resistance model is

\frac{dv}{dt}=9-\frac{v^2}{25}, \qquad v(0)=0, \qquad 0\le v<15

I.

Show that the terminal velocity predicted by the model is $15$ m s $^{-1}$ .

[1]

II.

Solve the differential equation exactly.

[4]

Use the exact solution to find the time when $v=12$ .

[2]

I.

Use Euler's method with $h=0.5$ to approximate $v(1)$ .

[2]

II.

Determine whether this Euler approximation is an overestimate or an underestimate. Justify your answer.

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 49

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A pollutant enters a lake after a spill. The amount of pollutant in the lake, $A$ kg, after $t$ days is modelled by

\frac{dA}{dt}+0.15A=60e^{-0.05t}, \qquad A(0)=0

The term $60e^{-0.05t}$ models the decreasing rate at which pollutant enters the lake.

I.

Find an integrating factor for the differential equation.

[1]

II.

Solve for $A$ in terms of $t$ .

[4]

Find the maximum amount of pollutant in the lake.

[4]

I.

Find the limiting amount of pollutant as $t\to\infty$ .

[1]

II.

Explain why the amount eventually decreases even though pollutant is still entering the lake.

[1]

Question 50

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $x>0$ , a curve satisfies

\frac{dy}{dx}=\frac{x+y}{x-y}, \qquad y(1)=0

Use the substitution $y=vx$ where appropriate.

I.

Show that the differential equation may be written as a separable differential equation in $v$ and $x$ .

[3]

II.

Hence find an implicit equation for the curve in terms of $x$ and $y$ .

[5]

I.

Find the equation of the tangent to the curve at $(1,0)$ .

[1]

II.

Using the implicit relation, determine the limiting value of $x$ as the solution approaches the line $y=x$ .

[2]

Question 51

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A tank contains $A$ grams of salt after $t$ minutes. Initially $A=12$ . For $0\le t\le4$ the amount of salt satisfies

\frac{dA}{dt}=8-\frac{A}{6}

After $t=4$ , fresh water is added so that, for $t>4$ ,

\frac{dA}{dt}=-\frac{A}{6}

Find $A$ in terms of $t$ for $0\le t\le4$ .

[5]

I.

Find the amount of salt in the tank when $t=4$ .

[2]

II.

Find $A$ in terms of $t$ for $t>4$ .

[3]

Find the elapsed time after $t=4$ until the amount of salt has returned to $12$ grams.

[2]

Question 52

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A curve satisfies

\frac{dy}{dx}=xy+\frac{x}{y}, \qquad y(0)=2

Let $z=y^2$ .

I.

Show that $z$ satisfies the linear differential equation $z'-2xz=2x$ .

[3]

II.

Hence solve the original differential equation, giving $y$ in terms of $x$ .

[5]

I.

Find the stationary point of the curve.

[2]

II.

Determine whether this stationary point is a local maximum or local minimum.

[1]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 53

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $x>0$ and $y>0$ , a curve satisfies

\frac{dy}{dx}=\frac{x+2y}{2x+y}

and passes through the point $(1,2)$ .

Use the substitution $y=vx$ to show that

x\frac{dv}{dx}=\frac{1-v^2}{v+2}

[3]

Hence show that the solution may be written in the form

\frac{v+1}{(v-1)^3}=3x^2

where $v=\frac{y}{x}$ .

[4]

I.

Determine the value of $y$ when $x=2$ .

[2]

II.

Determine the value of $y$ when $x=0.5$ .

[2]

Question 54

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $x>0$ , a curve satisfies

\frac{dy}{dx}=\frac{y}{x}+\left(\frac{y}{x}\right)^3, \qquad y(1)=1

Use the substitution $y=vx$ to obtain a separable differential equation in $v$ and $x$ .

[3]

Hence show that

y=\frac{x}{\sqrt{1-2\ln x}}

[3]

I.

Use Euler's method with step length $h=0.1$ to approximate $y(1.3)$ .

[2]

II.

Find the exact value of $y(1.3)$ and determine the largest possible value of $x$ for this solution.

[3]

Question 55

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A tank contains a salt solution. Let $A$ be the mass of salt, in kg, in the tank after $t$ minutes. A model for $A$ is

\frac{dA}{dt}+0.05A=0.4+0.1\sin(0.1t), \qquad A(0)=10

Solve the differential equation.

[4]

Find $A(30)$ .

[2]

I.

Determine the first time after $t=0$ at which $A=9$ .

[2]

II.

State the mean value and amplitude of the long-term oscillation of $A$ .

[2]

Question 56

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The velocity $v$ , in metres per second, of an object moving vertically through a fluid is modelled by

\frac{dv}{dt}=12-0.08v^2, \qquad v(0)=0

where $t$ is measured in seconds.

I.

Find the terminal velocity predicted by the model.

[2]

II.

Use Euler's method with step length $h=0.5$ to approximate $v(1.5)$ .

[2]

Solve the differential equation, giving $v$ in terms of $t$ .

[4]

I.

Find the time at which the object first reaches $90\%$ of its terminal velocity.

[2]

II.

The displacement from the starting point is denoted by $s$ . Given that $s(0)=0$ and $\frac{ds}{dt}=v$ , find the displacement when the object first reaches $90\%$ of its terminal velocity.

[2]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A curve which enters the first quadrant through $(1,0)$ satisfies

\frac{dy}{dx}=\frac{x+2y}{2x+y}, \qquad y(1)=0

The gradient at a point depends only on the ratio $\frac{y}{x}$ .

I.

Use the substitution $y=vx$ to obtain a separable differential equation in $v$ and $x$ .

[2]

II.

Show that

\frac{2+v}{1-v^2}=\frac{3}{2(1-v)}+\frac{1}{2(1+v)}

[2]

Hence solve the differential equation, giving an implicit equation in $x$ and $y$ .

[4]

I.

Find the point on the curve where $y=\frac{x}{2}$ . If you did not obtain an implicit equation, use $\frac{1+v}{(1-v)^3}=x^2$ .

[2]

II.

Deduce the limiting value of $\frac{y}{x}$ as $x$ increases along this branch of the curve.

[2]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For a positive constant $a$ , the function $y$ satisfies

y'+ay=e^{-x}, \qquad y(0)=0

The behaviour changes when $a=1$ .

I.

For $a\ne 1$ , solve the differential equation in terms of $a$ and $x$ .

[3]

II.

Solve the differential equation for the special case $a=1$ .

[2]

Show that the expression found in part (a)(i) tends to the result in part (a)(ii) as $a\to 1$ .

[2]

I.

For $a=2$ , find the maximum value of $y$ .

[3]

II.

Explain why $y>0$ for all $x>0$ when $a>0$ .

[1]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For a real constant $c$ , consider the family of solutions of

\frac{dy}{dx}+\frac{2x}{1+x^2}y=4x, \qquad y(0)=c

where $x\ge 0$ .

I.

Find an integrating factor for the differential equation.

[2]

II.

Show that the solution is

y=\frac{x^4+2x^2+c}{1+x^2}

[3]

Determine the value of $c$ for which the solution has a stationary point at $x=1$ .

[3]

I.

For $c=5$ , show that the stationary point at $x=1$ is a local minimum.

[3]

II.

Find the coordinates of this local minimum.

[1]

Question 60

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $x>0$ , a curve satisfies

\frac{dy}{dx}=\frac{x+y}{x-y}, \qquad y(1)=0

Use the substitution $y=vx$ to show that

x\frac{dv}{dx}=\frac{1+v^2}{1-v}

[3]

Hence show that the solution satisfies

\arctan\left(\frac{y}{x}\right)-\frac12\ln\left(1+\frac{y^2}{x^2}\right)=\ln x

[3]

I.

Determine the value of $y$ when $x=1.2$ .

[2]

II.

Deduce the limiting upper bound (supremum) of $x$ for this solution curve as it approaches the singular line $y=x$ .

[3]

Build a Practice Exam — Math AA HL

Test your knowledge with a custom paper containing questions from selected topics.

Differential Calculus

Integral Calculus

The Ultimate All-in-One
IB Platform

All content on this website has been developed independently from and is not endorsed by the International Baccalaureate Organization. International Baccalaureate and IB are registered trademarks owned by the International Baccalaureate Organization.