RevisionDojo

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

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 2

Consider the differential equation $z^2 \frac{\mathrm{d}w}{\mathrm{d}z} = w^2 - 2z^2$ for $z > 0$ and $w > 2z$ .

It is given that $w = 3$ when $z = 1$ .

Use Euler's technique, with a step length of 0.1, to find an approximate value of $w$ when $z = 1.5$ .

[4]

Use the substitution $w = vz$ to show that $z \frac{\mathrm{d}v}{\mathrm{d}z} = v^2 - v - 2$ .

[3]

By solving the differential equation, show that $w = \frac{8z + z^4}{4 - z^3}$ .

[8]

Find the actual value of $w$ when $z = 1.5$ .

[1]

Using the graph of $w = \frac{8z + z^4}{4 - z^3}$ , suggest a reason why the approximation given by Euler's technique in part is not a good estimate to the actual value of $w$ at $z = 1.5$ .

[1]

Question 5

HLPaper 2

Consider the differential equation

$\frac{dy}{dx} = g\left(\frac{y}{x}\right), \quad x > 0$

The curve $y = g(x)$ for $x > 0$ has a gradient function given by

$\frac{dy}{dx} = \frac{y^2 + 3xy + 2x^2}{x^2}$

The curve passes through the point $(1, -1)$ .

Use the substitution $y=wx$ to show that

$\int \frac{dw}{g(w)-w}=\ln x+D$

where $D$ is an arbitrary constant.

[3]

By using the result from part (a) or otherwise, solve the differential equation and hence show that the curve has equation

$y = x \tan(\ln(x)) - 1$ .

[6]

Use the differential equation $\frac{dy}{dx} = \frac{y^2 + 3xy + 2x^2}{x^2}$ to show that the points of zero gradient on the curve lie on two straight lines of the form $y = mx$ where the values of $m$ are to be determined.

[4]

The curve has a point of inflexion at $x_2, y_2$ where $e^{-\frac{\pi}{2}} < x_2 < e^{\frac{\pi}{2}}$ . Determine the coordinates of this point of inflexion.

[6]

Question 6

HLPaper 2

Consider the differential equation $\frac{dv}{du}=\frac{v - u}{v + u}$ , where $u, v > 0$ . It is given that $v = 2$ when $u = 1$ .

Solve the differential equation, giving your answer in the form $g(u, v) = 0$ .

[9]

The graph of $v$ against $u$ has a local maximum between $u=2$ and $u=3$ . Determine the coordinates of this local maximum.

[4]

Show that there are no points of inflexion on the graph of $v$ against $u$ .

[4]

Question 7

HLPaper 2

A large reservoir initially contains pure water. Water containing salt begins to flow into the reservoir. The solution is kept uniform by stirring and leaves the reservoir through an outlet at its base. Let $x$ grams represent the amount of salt in the reservoir and let $t$ minutes represent the time since the salt water began flowing into the reservoir.

The rate of change of the amount of salt in the reservoir, $\frac{{dx}}{{dt}}$ , is described by the differential equation $\frac{{dx}}{{dt}} = 10e^{-\frac{t}{4}} - \frac{x}{{t + 1}}$ .

Show that $t + 1$ is an integrating factor for this differential equation.

[2]

Hence, by solving this differential equation, show that ${x(t) = \frac{{200 - 40e^{-\frac{t}{4}}(t + 5)}}{{t + 1}}}$ .

[8]

Find the value of $t$ at which the amount of salt in the reservoir is decreasing most rapidly.

[2]

The rate of change of the amount of salt leaving the reservoir is equal to ${\frac{x}{{t + 1}}}$ .

Find the amount of salt that left the reservoir during the first 60 minutes.

[4]

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