AHL 5.18—Eulers method for 2nd order DEs Questionbank

Practice AHL 5.18—Eulers method for 2nd 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

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 of the object. The following model was proposed, where the object’s displacement, $x$ , from the top of the tube, measured in metres, is given by the differential equation

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

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 occasions and its terminal velocity recorded. It is found that the terminal velocity was consistently smaller than that predicted by the model used. It was suggested that the resistance to motion is actually proportional to the velocity squared and so the following model was set up:

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

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 the object 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 object predicted by this model.

[2]

Write down the differential equation for the second model 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 of the 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 exact terminal velocity for the object predicted by the second model.

[2]

Use your answers to parts 4, 5 and 6 to comment on the accuracy of the Euler approximation to this model.

[2]

Question 2

HLPaper 2

A mechanical engineer is analyzing the behavior of a car's suspension system, which can be modeled by the following differential equation:

$\frac{d^2 x}{d t^2} + 3 \frac{d x}{d t} + 2 x = 0$

Assume initial conditions $x(0)=1$ and $v(0)=0$ .

Rewrite the second-order differential equation as a system of two first-order equations that represent the motion of the car's suspension.

[5]

Use Euler's method with a step size of $h=0.1$ to approximate the values of $x(0.2)$ and $v(0.2)$ , given the initial conditions $x(0)=1$ and $v(0)=0$ .

[7]

Explain what happens to the motion of the car's suspension system over time based on the results.

[3]

Question 3

HLPaper 2

A physics student is studying the motion of a pendulum, which can be modeled using differential equations. The equation of motion for a pendulum under the small angle approximation is given by:

$\frac{d^2 \theta}{dt^2} + 9.8\theta = 0$

where $\theta$ is the angular displacement in radians and $t$ is the time in seconds.

Convert this second-order equation into two first-order equations that describe the angular motion of the pendulum.

[4]

Use Euler's method with a step size of $h = 0.2$ seconds to approximate the values of $\theta(0.4)$ and $\omega(0.4)$ , given the initial conditions $\theta(0) = 0.1$ radians and $\omega(0) = 0$ rad/s.

[8]

What does the result suggest about the motion of the pendulum?

[3]

Question 4

HLPaper 2

A drone is programmed to maintain a stable altitude while hovering, and its vertical motion is described by the following differential equation:

$\frac{d^2 x}{d t^2}=-5 x$

Rewrite the second-order equation as a system of first-order equations.

[4]

Use Euler's method with step size $h=0.1$ to approximate $x(0.2)$ and $v(0.2)$ , given that $x(0)=1$ and $v(0)=0$ .

[8]

Question 5

HLPaper 1

Consider the second order differential equation

$$\ddot{x} + 4(\dot{x})^2 - 2t = 0$$

where $x$ is the displacement of a particle for $t \geq 0$.

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

[2]

When $t=0$, $x = \dot{x} = 0$.

Use Euler’s method with a step length of $0.1$ to find an estimate for the value of the displacement and velocity of the particle when $t=1$.

[4]

Question 6

HLPaper 2

An object is released at the top of a vertical tube filled with a thick viscous fluid. The initial differential equation models the displacement $x(t)$ (in metres) of the object from the top of the tube:

\frac{d^2 x}{d t^2}=9.81-0.9 \frac{d x}{d t}

This model is revised due to experimental observations, yielding:

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

Substitute $v=\frac{d x}{d t}$ into the revised equation and solve the resulting differential equation to express $v(t)$ as a function of time.

[7]

Use your answer from part 1 to find the terminal velocity of the object under the revised model.

[2]

Rewrite the revised differential equation as a system of two first-order equations.

[2]

Use Euler's method with a step size $h=0.2$ to estimate the object's displacement and velocity at $t=0.6$ , starting with initial conditions $x(0)=0$ and $v(0)=0$ .

[4]

Using further iterations of Euler's method, approximate the terminal velocity of the object to five significant figures.

[1]

Find the exact terminal velocity using the differential equation.

[2]

Compare the Euler method approximation with the exact value obtained in part 6. Comment on the accuracy of the approximation.

[2]

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

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

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

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

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