AHL 5.16—Eulers method for 1st order DEs Questionbank

Practice AHL 5.16—Eulers method for 1st order DEs with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

A tank contains a saline solution with concentration $S$ (in $\mathrm{g} / \mathrm{L}$ ) at time $t$ (in minutes). Water with salt at $2 \mathrm{~g} / \mathrm{L}$ flows in at $3 \mathrm{~L} / \mathrm{min}$ , and the solution flows out at $3 \mathrm{~L} / \min$ . The tank volume is 100 L . The differential equation is:

\frac{d S}{d t}=0.06-0.03 S

Initial concentration:

S=1

g/L at $t=0$ .

Solve the differential equation to find $S(t)$ .

[4]

Use Euler's method with $h=1$ minute to estimate $S$ after 2 minutes.

[3]

Sketch the solution curve and Euler's approximations for $0 \leq t \leq 5$ .

[3]

Determine the time when $S=1.5$ $\mathrm{g} / \mathrm{L}$ using the analytical solution.

[3]

Question 2

HLPaper 1

A researcher models the concentration of a chemical pollutant $C$ (in $\mathrm{mg} / \mathrm{L}$ ) in a lake over time $t$ (in months). The rate of change of the concentration is influenced by natural degradation and an external source, modeled by the differential equation:

\frac{d C}{d t}=-0.2 C+5

Initially, at $t=0$ , the concentration is $C=10 \mathrm{mg} / \mathrm{L}$ .

Solve the differential equation to find the general solution for $C$ in terms of $t$ .

[4]

Use the initial condition to find the particular solution and determine the concentration after 3 months.

[3]

Use Euler's method with a step size of 0.5 months to estimate the concentration after 1 month.

[3]

Sketch the solution curve for $0 \leq t \leq 5$ and the Euler's method approximations for the first two steps on the same graph.

[3]

The lake is considered safe for aquatic life when $C<15 \mathrm{mg} / \mathrm{L}$ . Using your particular solution, determine if the lake is safe after 10 months.

[2]

Question 3

HLPaper 1

A predator-prey system models rabbits $R$ and foxes $F$ (in hundreds) over time $t$ (in years):

\frac{d R}{d t}=2 R-0.02 R F, \quad \frac{d F}{d t}=0.01 R F-0.5 F

Initial populations:

\begin{aligned} & R=10 \\ & F=2 \end{aligned}

at $t=0$ .

Interpret the terms in both differential equations.

[4]

Use Euler's method with

h=0.1

years to estimate

R

and

F

after 0.2 years.

[4]

Find the non-zero equilibrium point.

[3]

Sketch the phase portrait trajectory for

\begin{aligned} & 0 \leq t \leq 0.2 \\ & 5 \leq R \leq 15 \\ & 1 \leq F \leq 3 \end{aligned}

[2]

Question 4

HLPaper 1

Consider the curve defined by $x^{2}+y^{2}=e^{x y}$ , where $y \neq 0$ . The slope of the curve is given by the differential equation derived from implicit differentiation. (d) Use implicit differentiation to show that $\frac{d y}{d x}=\frac{y e^{x y}-2 x}{2 y - x e^{x y}}$ . (b) At the point $(0,1)$ , use Euler's method with a step size of 0.1 to estimate $y$ when $x=0.2$ . [3] (c) Verify that $(0,1)$ lies on the curve. [1] (d) Sketch the curve and the Euler's method approximation near $(0,1)$ for $0 \leq x \leq 0.2$ . [3]

Use implicit differentiation to show that $\frac{d y}{d x}=\frac{y e^{x y}-2 x}{2 y-x e^{x y}}$

[4]

At the point $(0,1)$ , use Euler's method with a step size of 0.1 to estimate $y$ when $x=0.2$ .

[3]

Verify that $(0,1)$ lies on the curve.

[1]

Sketch the curve and the Euler's method approximation near $(0,1)$ for $0 \leq x \leq 0.2$ .

[3]

Question 5

HLPaper 2

An object is placed into the top of a long vertical tube, filled with a thick viscous fluid,at time $t = 0$ seconds.

Initially it is thought that the resistance of the fluid would be proportional to the velocity ofthe object. The following model was proposed, where the object’s displacement, $x$ , fromthe top of the tube, measured in metres, is given by the differential equation

$\frac{{{{\text{d}}^2}x}}{{{\text{d}}{t^2}}} = 9.81 - 0.9\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}}} \right)$ .

The maximum velocity approached by the object as it falls is known as the terminal velocity.

An experiment is performed in which the object is placed in the fluid on a number of occasionsand its terminal velocity recorded. It is found that the terminal velocity was consistently smallerthan that predicted by the model used. It was suggested that the resistance to motion is actuallyproportional to the velocity squared and so the following model was set up.

$\frac{{{{\text{d}}^2}x}}{{{\text{d}}{t^2}}} = 9.81 - 0.9{\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}}} \right)^2}$

At terminal velocity the acceleration of an object is equal to zero.

By substituting $v = \frac{{{\text{d}}x}}{{{\text{d}}t}}$ into the equation, find an expression for the velocity of theparticle at time $t$ . Give your answer in the form $v = f(t)$ .

[7]

From your solution to part 1, or otherwise, find the terminal velocity of the objectpredicted by this model.

[2]

Write down the differential equation as a system of first order differential equations.

[2]

Use Euler’s method, with a step length of 0.2, to find the displacement and velocity ofthe object when $t = 0.6$ .

[4]

By repeated application of Euler’s method, find an approximation for the terminal velocity,to five significant figures.

[1]

Use the differential equation to find the terminal velocity for the object.

[2]

Use your answers to parts (d), (e) and (f) to comment on the accuracy of the Eulerapproximation to this model.

[2]

Question 6

HLPaper 2

A wildlife conservation project is monitoring the population of a certain species of bird over time. The population $y(t)$ changes according to the differential equation:

$\frac{d y}{d t} = 3y - 4t, \quad y(0) = 2$

Using Euler's method with a step size $h = 0.2$ , approximate the population $y(0.4)$ at time $t = 0.4$ .

[6]

Comment on whether using a smaller step size, such as $h = 0.1$ , would lead to a more accurate solution.

[3]

Explain one disadvantage of using a very small step size in Euler's method.

[2]

Question 7

HLPaper 3

This question will investigate the solution to a coupled system of differential equations and how to transform it to a system that can be solved by the eigenvector method.

It is desired to solve the coupled system of differential equations

$\dot x = x + 2y - 50$

$\dot y = 2x + y - 40$

where $x$ and $y$ represent the population of two types of symbiotic coral and $t$ is time measured in decades.

Find the equilibrium point for this system.

[2]

If initially $x = 100$ and $y = 50$ use Euler’s method with an time increment of 0.1 to find an approximation for the values of $x$ and $y$ when $t = 1$ .

[3]

Extend this method to conjecture the limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

Show how using the substitution $X = x - 10{\text{,}}\,\,Y = y - 20$ transforms the system of differential equations into $\begin{array}{*{20}{c}} {\dot X = X + 2Y} \\ {\dot Y = 2X + Y} \end{array}$ .

[3]

Solve this system of equations by the eigenvalue method and hence find the general solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system.

[8]

Find the particular solution to the original system, given the initial conditions of part (b).

[2]

Hence find the exact values of $x$ and $y$ when $t = 1$ , giving the answers to 4 significant figures.

[2]

Use part (f) to find limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

With the initial conditions as given in part (b) state if the equilibrium point is stable or unstable.

[1]

10.

If instead the initial conditions were given as $x = 20$ and $y = 10$ , find the particular solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system, in this case.

[2]

11.

With the initial conditions as given in part (j), determine if the equilibrium point is stable or unstable.

[2]

Question 8

HLPaper 2

The temperature $T(t)$ of water in a tank changes according to the equation: $\frac{d T}{d t}=0.5(80-T), \quad T(0)=20$

Calculate the temperature $T(2)$ using Euler's method with a step size $h=1$ .

[4]

Solve the differential equation analytically to find the exact solution for $T(t)$ .

[4]

Determine the percentage error in your approximation from part (a).

[2]

Explain what the limiting temperature of the water is as $t \rightarrow \infty$ .

[1]

Question 9

HLPaper 2

A research team is investigating the dynamics between a prey species and its predator in a natural ecosystem. The interactions can be modeled by the following system of differential equations:

$\frac{d x}{d t}=2 x-0.01 x y, \quad \frac{d y}{d t}=-y+0.02 x y$

where $x(t)$ represents the prey population and $y(t)$ represents the predator population.

Using Euler's method with a step size of $h=0.1$ , compute the populations $x(0.1)$ and $y(0.1)$ given the initial conditions $x(0)=100$ and $y(0)=20$ .

[10]

Describe the behaviour of the two populations over time.

[6]

Suggest how a change in the parameter values might alter the dynamics of the system.

[4]

Question 10

HLPaper 2

A researcher is analyzing the growth of a bacterial culture, which can be modeled by the differential equation:

$\frac{d y}{d x} = 3y + 4, \quad y(0) = -1$

Use a step size $h = 0.2$ to approximate the bacterial population at $x = 0.4$ .

[8]

Discuss the limitations of Euler's method in comparison with other numerical methods like the Runge-Kutta method.

[4]

An object is placed into the top of a long vertical tube, filled with a thick viscous fluid,at time $t = 0$ seconds.

$\frac{{{{\text{d}}^2}x}}{{{\text{d}}{t^2}}} = 9.81 - 0.9\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}}} \right)$ .

The maximum velocity approached by the object as it falls is known as the terminal velocity.

$\frac{{{{\text{d}}^2}x}}{{{\text{d}}{t^2}}} = 9.81 - 0.9{\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}}} \right)^2}$

At terminal velocity the acceleration of an object is equal to zero.