IB Mathematics Analysis and Approaches Topic AHL 5.18—1st Order De’s – Euler Method, Variables Separable, Integrating Factor, Homogeneous DE Using Sub Y=vx Questionbank (HL)

IB Mathematics Analysis and Approaches Topic AHL 5.18—1st Order De’s – Euler Method, Variables Separable, Integrating Factor, Homogeneous DE Using Sub Y=vx Questionbank (HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic AHL 5.18—1st Order De’s – Euler Method, Variables Separable, Integrating Factor, Homogeneous DE Using Sub Y=vx with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 5.18—1st Order De’s – Euler Method, Variables Separable, Integrating Factor, Homogeneous DE Using Sub Y=vx and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

A population $P$ of bacteria grows according to the differential equation

$\frac{d P}{d t} = \frac{P ( 1000 - P )}{1000}, t \geq 0$

where $t$ is time in hours and $P$ is the population size. At $t = 0, P = 200$ .

Solve the differential equation, given $P (0) = 200$ , to find $P (t)$ .

[8]

Find the exact time $t$ when the population reaches $500$ .

[4]

Question 2

HLPaper 2

Consider the differential equation

$\frac{d y}{d x} = \frac{y ^{2} - x ^{2}}{x y}, x, y > 0$

with the initial condition $y = 2$ when $x = 1$ .

Use Euler's method with step size $h = 0.2$ to estimate $y$ at $x = 1.4$ , giving your answer in exact form.

[5]

Solve the differential equation using the substitution $y = v x$ , giving your answer in the form $y = f (x)$ .

[8]

Sketch the solution curve and the isocline for $\frac{d y}{d x} = 0$ for $0 < x \leq e^{2}$ .

[3]

Question 3

HLPaper 1

Consider the differential equation

$\frac{d y}{d x} + \frac{y}{x} = sin x, x > 0$

with the initial condition $y = 0$ when $x = \frac{π}{2}$ .

Find the general solution using the integrating factor method.

[6]

Determine the particular solution satisfying the initial condition.

[3]

Question 4

HLPaper 1

Consider the differential equation

$\frac{d y}{d x} = \frac{2 x + y}{x}, x > 0,$

with the initial condition $y = 0$ when $x = 1$ .

Solve the differential equation, giving your answer in the form $y = f (x)$ .

[7]

Find the coordinates of the point where the solution curve intersects the line $y = x$ .

[3]

Determine whether the solution curve has any points of inflection.

[4]

Question 5

HLPaper 1

Consider the differential equation

$\frac{d y}{d x} + \frac{2 y}{x} = e^{x}, x > 0$

with the initial condition $y = 1$ when $x = 1$ .

Show that $x^{2}$ is an integrating factor for this differential equation.

[3]

Solve the differential equation, giving your answer in the form $y = f (x)$ .

[7]

IB Mathematics Analysis and Approaches AHL 5.18—1st Order De’s – Euler Method, Variables Separable, Integrating Factor, Homogeneous DE Using Sub Y=vx Questionbank

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IB Mathematics Analysis and Approaches AHL 5.18—1st Order De’s – Euler Method, Variables Separable, Integrating Factor, Homogeneous DE Using Sub Y=vx Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 5.18—1st Order De’s – Euler Method, Variables Separable, Integrating Factor, Homogeneous DE Using Sub Y=vx Questionbank

Unlock all 26 questions with a free account