Past Exam Questions

The variables x and y satisfy the differential equation \(\frac{dy}{dx} = ky^3 e^{-x}\), where \(k\) is a constant. It is given that \(y = 1\) when \(x = 0\), and that \(y = \sqrt{e}\) when \(x = 1\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

The variables x and θ satisfy the differential equation

\(x \cos^2 \theta \frac{dx}{d\theta} = 2 \tan \theta + 1,\)

for \(0 \leq \theta < \frac{1}{2}\pi\) and \(x > 0\). It is given that \(x = 1\) when \(\theta = \frac{1}{4}\pi\).

(i) Show that \(\frac{d}{d\theta}(\tan^2 \theta) = \frac{2 \tan \theta}{\cos^2 \theta}\).

(ii) Solve the differential equation and calculate the value of \(x\) when \(\theta = \frac{1}{3}\pi\), giving your answer correct to 3 significant figures.

The variables x and y satisfy the differential equation \(\frac{dy}{dx} = 4 \cos^2 y \tan x\), for \(0 \leq x < \frac{1}{2}\pi\), and \(x = 0\) when \(y = \frac{1}{4}\pi\). Solve this differential equation and find the value of \(x\) when \(y = \frac{1}{3}\pi\).

The variables x and y satisfy the differential equation

\(\frac{dy}{dx} = e^{-2y} \tan^2 x\),

for \(0 \leq x < \frac{1}{2}\pi\), and it is given that \(y = 0\) when \(x = 0\). Solve the differential equation and calculate the value of \(y\) when \(x = \frac{1}{4}\pi\).

The variables x and θ satisfy the differential equation

\((3 + \\cos 2\theta) \frac{dx}{d\theta} = x \sin 2\theta,\)

and it is given that \(x = 3\) when \(\theta = \frac{1}{4}\pi.\)

(i) Solve the differential equation and obtain an expression for \(x\) in terms of \(\theta.\) [7]

(ii) State the least value taken by \(x.\) [1]

The variables x and θ satisfy the differential equation \(\frac{dx}{dθ} = (x + 2) \sin^2 2θ\), and it is given that \(x = 0\) when \(θ = 0\). Solve the differential equation and calculate the value of x when \(θ = \frac{1}{4}π\), giving your answer correct to 3 significant figures.

The variables x and θ satisfy the differential equation

\(2 \cos^2 \theta \frac{dx}{d\theta} = \sqrt{2x + 1}\),

and \(x = 0\) when \(\theta = \frac{1}{4}\pi\). Solve the differential equation and obtain an expression for \(x\) in terms of \(\theta\).

The variables x and θ satisfy the differential equation

\(\frac{x}{\tan \theta} \frac{\mathrm{d}x}{\mathrm{d}\theta} = x^2 + 3.\)

It is given that \(x = 1\) when \(\theta = 0\).

Solve the differential equation, obtaining an expression for \(x^2\) in terms of \(\theta\).

The variables x and y are related by the differential equation

\(\frac{dy}{dx} = \frac{6ye^{3x}}{2 + e^{3x}}\).

Given that \(y = 36\) when \(x = 0\), find an expression for \(y\) in terms of \(x\).

The variables x and y satisfy the differential equation

\(\frac{dy}{dx} = e^{2x+y}\),

and \(y = 0\) when \(x = 0\). Solve the differential equation, obtaining an expression for y in terms of x.

The variables x and θ are related by the differential equation

\(\sin 2θ \frac{dx}{dθ} = (x + 1) \cos 2θ\),

where \(0 < θ < \frac{1}{2}π\). When \(θ = \frac{1}{12}π\), \(x = 0\). Solve the differential equation, obtaining an expression for \(x\) in terms of \(θ\), and simplifying your answer as far as possible.

The variables x and t are related by the differential equation

\(e^{2t} \frac{dx}{dt} = \cos^2 x\),

where \(t \geq 0\). When \(t = 0\), \(x = 0\).

(i) Solve the differential equation, obtaining an expression for \(x\) in terms of \(t\). [6]

(ii) State what happens to the value of \(x\) when \(t\) becomes very large. [1]

(iii) Explain why \(x\) increases as \(t\) increases. [1]

(a) The variables x and y satisfy the differential equation \(\frac{dy}{dx} = \frac{4 + 9y^2}{e^{2x+1}}\).

It is given that \(y = 0\) when \(x = 1\).

Solve the differential equation, obtaining an expression for y in terms of x.

(b) State what happens to the value of y as x tends to infinity. Give your answer in an exact form.

The variables x and y satisfy the differential equation

\(\cos 2x \frac{dy}{dx} = \frac{4 \tan 2x}{\sin^2 3y}\),

where \(0 \leq x < \frac{1}{4}\pi\). It is given that \(y = 0\) when \(x = \frac{1}{6}\pi\).

Solve the differential equation to obtain the value of x when \(y = \frac{1}{6}\pi\). Give your answer correct to 3 decimal places.

The variables x and y satisfy the differential equation

\(\frac{dy}{dx} = e^{3y} \sin^2 2x\).

It is given that \(y = 0\) when \(x = 0\).

Solve the differential equation and find the value of \(y\) when \(x = \frac{1}{2}\).

The variables x and θ satisfy the differential equation

\(x \sin^2 \theta \frac{dx}{d\theta} = \tan^2 \theta - 2 \cot \theta,\)

for \(0 < \theta < \frac{1}{2}\pi\) and \(x > 0\). It is given that \(x = 2\) when \(\theta = \frac{1}{4}\pi\).

(a) Show that \(\frac{d}{d\theta}(\cot^2 \theta) = -\frac{2 \cot \theta}{\sin^2 \theta}\).

(You may assume without proof that the derivative of \(\cot \theta\) with respect to \(\theta\) is \(-\csc^2 \theta\).) [1]

(b) Solve the differential equation and find the value of \(x\) when \(\theta = \frac{1}{6}\pi\). [7]

(a) Given that \(y = \ln(\ln x)\), show that \(\frac{dy}{dx} = \frac{1}{x \ln x}\).

The variables \(x\) and \(t\) satisfy the differential equation \(x \ln x + t \frac{dx}{dt} = 0\).

It is given that \(x = e\) when \(t = 2\).

(b) Solve the differential equation obtaining an expression for \(x\) in terms of \(t\), simplifying your answer.

(c) Hence state what happens to the value of \(x\) as \(t\) tends to infinity.

The variables x and y satisfy the differential equation

\((1 - \\cos x) \frac{dy}{dx} = y \sin x.\)

It is given that \(y = 4\) when \(x = \pi.\)

(a) Solve the differential equation, obtaining an expression for \(y\) in terms of \(x.\) [6]

(b) Sketch the graph of \(y\) against \(x\) for \(0 < x < 2\pi.\) [1]

The variables x and t satisfy the differential equation

\(e^{3t} \frac{dx}{dt} = \cos^2 2x\),

for \(t \geq 0\). It is given that \(x = 0\) when \(t = 0\).

(a) Solve the differential equation and obtain an expression for \(x\) in terms of \(t\). [7]

(b) State what happens to the value of \(x\) when \(t\) tends to infinity. [1]

The diagram shows the graphs with equations \(y = f(x)\) and \(y = g(x)\).

Describe fully a sequence of two transformations which transforms the graph of \(y = f(x)\) to the graph of \(y = g(x)\). Make clear the order in which the transformations should be applied.