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

The motion of a particle is given by:

\ddot{x}+5 \dot{x}^{2}=2 \sin (t),

with $x$ in metres, $t \geq 0$ .

Convert to a system of first-order differential equations.

[2]

With $x(0)=1, \dot{x}(0)=-1$ , use Euler's method with step size 0.2 to estimate $x$ and $\dot{x}$ at $t=0.6$ .

[4]

Estimate the acceleration at $t=0.6$ using part (b).

[2]

Suggest one limitation of Euler's method in this context.

[2]

Question 2

HLPaper 2

The motion of a particle is modeled by:

\ddot{x}=3 t^{2}-2 \dot{x}^{2}+x,

with $x$ in metres, $t \geq 0$ .

Convert the equation into a system of first-order differential equations.

[2]

With $x(0)=1, \dot{x}(0)=0$ , use Euler's method with step size 0.2 to estimate $x$ and $\dot{x}$ at $t=0.6$ .

[4]

Estimate the rate of change of velocity at $t=0.6$ using part (b).

[2]

Comment on the suitability of the step size used in part (b) for this differential equation.

[2]

Question 3

HLPaper 2

A particle's motion is described by:

\ddot{x}+4 \dot{x}^{2}=e^{-2 t},

with $x$ in metres, $t \geq 0$ .

Express as a system of first-order differential equations.

[2]

With $x(0)=0, \dot{x}(0)=1$ , use Euler's method with step size 0.2 to estimate $x$ and $\dot{x}$ at $t=0.4$ .

[4]

Estimate the acceleration at $t=0.4$ using part (b).

[2]

Suggest why Euler's method might overestimate or underestimate the velocity at $t=0.4$ .

[2]

Question 4

HLPaper 1

A spring system is governed by:

\ddot{x}+3 \dot{x}+2 x=\cos (2 t),

with $x$ in metres, $t \geq 0$ .

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

[2]

With $x(0)=0, \dot{x}(0)=0$ , use Euler's method with step size 0.1 to estimate $x$ and $\dot{x}$ at $t=0.5$ .

[4]

Sketch the velocity $\dot{x}$ against $t$ from $t=0$ to $t=0.5$ .

[3]

Question 5

HLPaper 1

A particle's motion is described by:

\ddot{x}=2 t-5 \dot{x}+x,

with $x$ in metres, $t \geq 0$ .

Convert the equation into a system of first-order differential equations.

[2]

Given $x(0)=0, \dot{x}(0)=2$ , use Euler's method with step size 0.1 to find $x$ and $\dot{x}$ at $t=0.3$ .

[4]

Sketch the velocity $\dot{x}$ against $t$ from $t=0$ to $t=0.3$ .

[3]

Question 6

HLPaper 2

The motion of a spring system is given by:

\ddot{x}+2 \dot{x}^{2}-4 x=t,

with $x$ in metres, $t \geq 0$ .

Write as a system of first-order differential equations.

[2]

With $x(0)=1, \dot{x}(0)=-1$ , use Euler's method with step size 0.1 to estimate $x$ and $\dot{x}$ at $t=0.3$ .

[4]

Estimate the acceleration at $t=0.3$ .

[2]

Question 7

HLPaper 2

A system is modeled by:

\ddot{x}+4 \dot{x}^{2}=t^{2},

with $x$ in metres, $t \geq 0$ .

Write as a system of first-order differential equations.

[2]

With $x(0)=0, \dot{x}(0)=0$ , use Euler's method with step size 0.1 to estimate $x$ and $\dot{x}$ at $t=0.2$ .

[3]

Estimate the acceleration at $t=0.2$ using part (b).

[2]

Question 8

HLPaper 1

A particle moves according to:

\ddot{x}+6 \dot{x}+9 x=3 t,

with $x$ in metres, $t \geq 0$ .

Convert to a system of first-order differential equations.

[2]

With $x(0)=0, \dot{x}(0)=0$ , use Euler's method with step size 0.2 to find $x$ and $\dot{x}$ at $t=0.6$ .

[4]

Sketch the displacement $x$ against $t$ from $t=0$ to $t=0.6$ .

[3]

Question 9

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 10

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]

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.