Differential Equations (OCR A Level Maths A: Pure): Exam Questions

Find the general solution to the differential equation

$\frac{dy}{dx}={\sin }^2\left(2y\right)$

giving your answer in the form $x=\mathrm{f}\left(y\right)+c$ where $c$ is a constant and $\mathrm{f}\left(y\right)$ is a function to be found.

Find the general solution to the differential equation

$9t^2-4+\frac{dx}{dt}=0$

Find the particular solution to the differential equation

$\frac{dV}{dx}-4=2{\mathrm{e}}^x$

given that the graph of $V$ against $x$ passes through the point with coordinates $\left(0,3\right)$.

A differential equation is given by

${\mathrm{e}}^{-3x}\frac{dy}{dx}=2{\mathrm{e}}^y$

It is known that $y=0$ when $x=0$.

Solve the differential equation, giving your answer in the form

$p{\mathrm{e}}^{3x}+{\mathrm{e}}^{-y}=q$

where $p$ and $q$ are rational numbers to be found.

A differential equation is given by

${\sin }^{2}x\frac{dy}{dx}={\cos }^{2}y$

where $y=0$ when $x=\frac{\pi }{4}$.

Solve the differential equation, giving your answer in the form

$\tan y=\mathrm{f}\left(x\right)$

where $\mathrm{f}\left(x\right)$ is a function to be found.

A weather balloon of volume $V$ m³ is being inflated, where $t$ is the time in minutes after inflation begins.

• The rate of change of its volume is inversely proportional to its volume

• When the rate of inflation of the balloon is 10 m³ min^-1, the volume of the balloon is 20 m³

Use this information to write down a suitable differential equation for $V$ and $t$.

Show that the general solution to the differential equation is

$V^2=400t+c$

where $c$ is a constant.

Initially, the balloon is flat with a volume of 0 m³.

Find the volume of the balloon after 25 minutes.

A disease affecting trees is spreading throughout a large forested area. Let $N$ be the number of infected trees $t$ days after the disease was first discovered.

A model for $N$ and $t$ is given by

$\frac{dN}{dt}=kNt$

where $k$ is a positive constant.

It is known that

• When the disease was first discovered, 3 trees were infected

• Ten days after the disease was first discovered, 10 trees were infected

Solve the differential equation to show that

$N=3{\mathrm{e}}^{a{t}^2}$

where

$a=\frac{1}{100}\ln \left(\frac{10}{3}\right)$

Scientists believe the majority of the forest can be saved from infection if action is taken before 30 trees are infected.

Find the number of days (since first discovering the disease) that the model predicts scientists have in order to take action.

Find the general solution to the differential equation

$\frac{2y-1}{3}\frac{dy}{dx}=x^2{y}^2-x^{2}y$

where $y>1$, giving your answer in the form

$y^2-y=\mathrm{f}\left(x\right)$

Find the general solution to the differential equation

$3\frac{dy}{dx}=\frac{\mathrm{cosec}{y}^3}{y^2}$

giving your answer in the form $x=\mathrm{g}\left(y\right)$.

A hot air balloon is being inflated at a rate that is inversely proportional to the square of its volume.

Defining variables for the volume of the balloon (m³) and time (seconds), write down a differential equation to describe the relationship between volume and time as the hot air balloon is inflated.

You are given the following information:

• Initially, the hot air balloon has a volume of zero

• After 400 seconds of inflating, its volume is 600 m³

• The hot air balloon is considered ready for release when its volume reaches 1250 m³

If the hot air balloon needs to be ready for release by midday, find the latest time that it can start being inflated.

Find the general solution to the differential equation

$\frac{1}{2}{\sec }^2\left(3t\right)+2\frac{dx}{dt}=0$

Find the particular solution to the differential equation

$2x{\mathrm{e}}^{4x}-3\frac{dV}{dx}=1$

where the graph of $V$ against $x$ passes through the point with coordinates $\left(0,2\right)$.

Find the general solution to the differential equation

$\frac{dy}{dx}=2xy+2x-y-1$

where $y>-1$, giving your answer in the form $y=\mathrm{f}\left(x\right)$.

Palm trees are being planted on an island. Let $N$ be the total number of palm trees planted on the island after $t$ days.

The variables $N$ and $t$ are modelled by the differential equation

$\frac{\mathrm{d}N}{\mathrm{d}t}=kN\left(N-1\right),N>1$

where $N>1$ and $k$ is a positive constant.

By solving the differential equation, show that

$N=\frac{1}{1-A{\mathrm{e}}^{kt}}$

where $A$ is a positive constant.

It is known that

• Initially 2 palm trees are planted

• After 14 days, 4 palm trees in total have been planted

Use this information to show that

$k=\frac{1}{14}\ln p$

where $p$ is a rational number to be found.

By considering the form of the solution to the differential equation, suggest a range of values of $t$ for which the model is valid.

The temperature of a heated object, $T$°C, cools over time, $t$ minutes. The room temperature (called the ambient temperature) is constant, $T_{\mathrm{amb}}$, where $T>T_{\mathrm{amb}}$.

Newton’s Law of Cooling states that the rate of decrease in temperature of a heated object is directly proportional to the difference between the object’s temperature and the ambient temperature.

By forming and solving a differential equation in $T$ and $t$ (involving the constant $T_{\mathrm{amb}}$ and a positive constant of proportionality, $k$) show that

$T=T_{\mathrm{amb}}+A{\mathrm{e}}^{-kt}$

For food safety reasons, a meat processing factory must store its products at a temperature of below -1 °C.

• One particular product has a temperature of 7 °C

• It is placed in one of the factory's freezers, which has a constant ambient temperature of -4 °C

• One minute later, its temperature has dropped to 4.7 °C.

• Any products that fail to cool to below -1 °C within 6 minutes must be discarded

Determine whether or not this product will need to be discarded.

Show that the solution to the differential equation

$\cos x\frac{dy}{dx}=\cos y$

where $y=\pi$ when $x=0$ may be written in the form

$\left|\tan \left(\frac{y}{2}+\frac{\pi }{4}\right)\right|=\left|\tan \left(\frac{x}{2}+\frac{\pi }{4}\right)\right|$

(i) Prove that if $\left|\tan \left(\frac{y}{2}+\frac{\pi }{4}\right)\right|=\left|\tan \left(\frac{x}{2}+\frac{\pi }{4}\right)\right|$ then

$y=x+2n\pi \mathrm{or}y=-x+\left(2n-1\right)\pi$

where $n$ is an integer.

(ii) Hence deduce that the particular solution to the differential equation in part (a) is

$y=\pi -x$