AHL 5.18—1st order DE’s – Euler method, variables separable, integrating factor, homogeneous DE using sub y=vx Questionbank

Practice AHL 5.18—1st order DE’s – Euler method, variables separable, integrating factor, homogeneous DE using sub y=vx with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Consider the homogeneous differential equation $\frac{dy}{dx} = \frac{y^2-2x^2}{xy}$ , where $x, y \neq 0$ . It is given that $y = 2$ when $x = 1$ .

By using the substitution $y = vx$ , solve the differential equation. Give your answer in the form $y^2 = f(x)$ .

[8]

The points of zero gradient on the curve $y^2 = f(x)$ lie on two straight lines of the form $y = mx$ where $m \in \mathbb{R}$ . Find the values of $m$ .

[2]

Question 2

HLPaper 1

Consider the differential equation $\frac{dy}{dx} = \frac{x^2 + y^2}{xy}$ .

By using the substitution $y = vx$ , show that the differential equation can be transformed into a separable form.

[3]

Solve the separable differential equation obtained in part 1.

[3]

Express the solution in terms of $y$ and $x$ .

[2]

Using Euler's method with a step size of 0.1, estimate the value of $y$ at $x = 1.1$ given that $y(1) = 1$ .

[2]

Question 3

HLPaper 1

This problem involves solving a first-order differential equation using the integrating factor method.

Consider the differential equation $\frac{dy}{dx} + y = e^x$ . Solve this differential equation using the integrating factor method.

[5]

Verify that the solution $y = \frac{1}{2} e^x + Ce^{-x}$ satisfies the original differential equation.

[3]

Using Euler's method with a step size of 0.1, estimate the value of $y$ at $x = 0.1$ given that $y(0) = 0$ .

[4]

Question 4

HLPaper 1

Consider the system of differential equations

\frac{\mathrm{d} x}{\mathrm{~d} t}=2 x-y, \quad \frac{\mathrm{~d} y}{\mathrm{~d} t}=x+2 y

where $x, y, t \in \mathbb{R}$ , with initial conditions $x=0, y=1$ at $t=0$ .

By differentiating the first equation and substituting, show that $x$ satisfies

\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-4 \frac{\mathrm{~d} x}{\mathrm{~d} t}+5 x=0

[4]

Solve the differential equation in part (a) to find $x$ as a function of $t$ .

[5]

Hence find $y$ as a function of $t$ .

[3]

Question 5

HLPaper 1

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

\frac{\mathrm{d} P}{\mathrm{~d} t}=\frac{P(1000-P)}{1000}, \quad t \geq 0

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

Solve the differential equation to find $P$ as a function of $t$ .

Sketch the solution curve, indicating the behavior as $t \rightarrow \infty$ .

Find the time $t$ when the population reaches 500 , correct to two decimal places.

[4]

Question 6

HLPaper 1

Consider the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y^{2}-x^{2}}{x y}, \quad x, y>0

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

Use Euler's method with step size $h=0.1$ to estimate $y$ at $x=1.2$ , correct to three decimal places.

[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 $\frac{\mathrm{d} y}{\mathrm{~d} x}=0$ .

[3]

Question 7

HLPaper 1

Consider the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{y}{x}=\sin x, \quad x>0

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

Find the general solution using the integrating factor method.

[6]

Determine the particular solution satisfying the initial condition.

[3]

Question 8

HLPaper 1

A tank initially contains 100 liters of pure water. A solution containing $0.2 \mathrm{~kg} / \mathrm{L}$ of salt enters at $5 \mathrm{~L} / \mathrm{min}$ , and the well-mixed solution exits at the same rate. Let $S$ (in kg ) be the amount of salt in the tank at time $t$ (in minutes).

Show that the differential equation modeling the amount of salt is

\frac{\mathrm{d} S}{\mathrm{~d} t}=1-\frac{S}{20} .

Solve the differential equation, given $S=0$ at $t=0$ , to find $S$ as a function of $t$ .

[7]

Sketch the solution curve for $0 \leq t \leq 60$ , indicating the asymptotic behavior. marks]

Question 9

HLPaper 1

Consider the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 y}{x}=e^{x}, \quad x>0

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

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

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

Find the value of $x$ where $y$ is maximized, for $x>0$ .

Question 10

HLPaper 2

Consider the differential equation $\frac{dy}{dx} = x \sqrt{y}, \quad \text{for } x \ge 0, y > 0.$ The initial condition is $y=4$ when $x=0$ .

Use Euler’s method with a step size of $h=0.2$ to approximate the value of $y$ when $x=0.4$ .

[4]

Find the exact solution of the differential equation in the form $y=g(x)$ .

[5]

Calculate the percentage error in the approximation found in part 1 compared to the exact value at $x=0.4$ .

[3]