IB Mathematics Applications & Interpretation Topic AHL 5.18—eulers Method for 2nd Order Des Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 5.18—eulers Method for 2nd Order Des Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 5.18—eulers Method for 2nd Order Des with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 5.18—eulers Method for 2nd Order Des and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 2

The motion of a particle is given by:

$\overset{x}{¨} + 5 \overset{x}{˙}^{2} = 2 sin (t),$

with $x$ in metres, $t$ in seconds $(t \geq 0)$ , and $sin (t)$ using radian measure.

Convert to a system of first-order differential equations.

[2]

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

[4]

Hence estimate the acceleration at $t = 0.6$ .

[2]

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

[2]

Question 2

HLPaper 2

A particle moves along a straight line. Its displacement $x$ (in metres) at time $t$ (in seconds), $t \geq 0$ , satisfies

$\overset{x}{¨} + 4 \overset{x}{˙}^{2} = e^{- 2 t} .$

Express this as a system of first-order differential equations.

[2]

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

[4]

Use your estimate of $\overset{x}{˙} (0.4)$ from Part 1 to estimate the acceleration at $t = 0.4$ .

[2]

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

[2]

Question 3

HLPaper 1

A spring system is governed by:

$\overset{x}{¨} + 3 \overset{x}{˙} + 2 x = cos (2 t),$

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

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

[2]

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

[4]

Hence, sketch the graph of $\overset{x}{˙}$ against $t$ for $0 \leq t \leq 0.5$ , using the Euler estimates from the previous part.

[3]

Question 4

HLPaper 1

A system is modeled by:

$\overset{x}{¨} + 2 \overset{x}{˙} + 3 x = t^{3},$

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

Write as a system of first-order differential equations.

[2]

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

[4]

Using Euler estimates at $t = 0, 0.1, 0.2, 0.3$ , sketch the displacement $x$ against $t$ for $0 \leq t \leq 0.3$ .

[3]

Question 5

HLPaper 1

A particle's motion is governed by the second-order differential equation

$\overset{x}{¨} + 3 \overset{x}{˙} + 2 x = sin (t),$

where $x$ is the displacement in metres at time $t$ seconds, $t \geq 0$ .

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

[2]

Given initial conditions $x (0) = 0, \overset{x}{˙} (0) = 1$ , use Euler's method with a step size of $0.1$ to estimate the displacement and velocity at $t = 0.5$ .

[4]

Sketch the approximate solution for displacement $x$ from $t = 0$ to $t = 0.5$ using the points calculated in Part 1.

[3]

IB Mathematics Applications & Interpretation AHL 5.18—eulers Method for 2nd Order Des Questionbank

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IB Mathematics Applications & Interpretation AHL 5.18—eulers Method for 2nd Order Des Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 5.18—eulers Method for 2nd Order Des Questionbank

Unlock all 19 questions with a free account