1st order Differential equations (separable method): notes and practice questions

If a tank of water is leaking through a hole at the bottom and the rate at which the volume of water $V\ \left( cm^{3} \right)$ changes with respect to time $t(seconds)$ is proportional to the volume itself, and is described by the differential equation:

$\frac{dV}{dt} = - \frac{V}{t}$

$a$ solve the differential equation to find the equation of $V(t)$ if after 5 seconds the volume of the water becomes $30,000\ cm^{3}$.

$b$ Find at what time the volume of the tank becomes $10,000\ cm^{3}$ according to the calculated model.

(a) A cylindrical grain silo with a radius of $2.5$ m and an initial grain height of $10$ m is being emptied. The rate at which the grain is removed from the silo is proportional to the square root of the height of the grain.

Using $V$ for the volume of grain in the silo and $h$ for the height of the grain, write down a differential equation relating these variables.

(b) Apply the chain rule and the formula for the volume of a cylinder to show that $\frac{dV}{dt} = 6.25\pi \frac{dh}{dt}$. Hence, write your answer to part (a) in terms of $h$ and $t$.

(c) After $30$ minutes, the height of the grain has dropped to $6.25$ m. Predict how long it will take for the silo to be completely empty from the initial height of $10$ m.

If a vehicle is moving along a straight road, where its position $y$ at time $t$ for $t \geq 0$ is given by the following differential equation:

$\frac{dy}{dt} = ysin(t + 1)$

At $t = 0$ take $y = 1$.

$a$ Use Euler's method with a step size of 0.1 and estimate $y$ at $t = 0.3$.

$b$ By solving the differential equation analytically, calculate the percentage error in your approximation from part (a) at $t = 0.3.$

A specialized chemical reactor is designed to produce a certain compound. The rate of change of the concentration of a key reactant, $C$, with respect to time, $t$, is modeled by the differential equation $\frac{dC}{dt} = \frac{t^2}{C}$.

At $t = 1$ minute, the concentration $C$ is $2$ mol/L.

(a) Use Euler's method with a step size of $0.1$ to find approximate values of $C$ when $t = 1.1$, $1.2$, and $1.3$ minutes. Give your answers to three decimal places.

(b) Solve the differential equation $\frac{dC}{dt} = \frac{t^2}{C}$ to find the exact concentration $C$ as a function of $t$. Hence, find the absolute errors for each of your approximations in part (a). Give your exact values to five decimal places and absolute errors to five decimal places.

If a transformation $S$ maps the vector $\begin{pmatrix} x \\ y \end{pmatrix}$ to $\begin{pmatrix} x' \\ y' \end{pmatrix}$ as follows:

$S:\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 4 & - 7 \\ 3 & 5 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} 2 \\ - 3 \end{pmatrix}$

$a$ If the image of a point $A(x,y)$, due to this transformation, has coordinates (0,16), Find the coordinates of point $A(x,y)$.

$b$ If a quadrilateral $M$ undergoes the transformation $S$, what happens to the area of its image?

A microbiologist is studying the interaction between two competing species of bacteria, species A and species B, in a controlled environment. Let $A$ represent the population of species A (measured in millions) and $B$ represent the population of species B (measured in millions). Let $t$ represent time, in hours.

The interaction can be modelled by the coupled differential equations:

$\frac{dA}{dt} = A(1 - B)$

$\frac{dB}{dt} = B(A - 3)$

State the two equilibrium points for this model.

Initially, there are $4$ million bacteria of species A and $0.5$ million bacteria of species B. Use the Euler method with a step size of $0.2$ hours to estimate the population of species A and species B after $1$ hour (to the nearest integer). Show the intermediate values that are obtained in the working, in the format of a table.

Suggest whether stating the population of bacteria to the nearest integer is a valid level of accuracy when using Euler's method in part (b).

A scientific experiment involves a platform that oscillates with damped motion. The displacement, $x$, of the platform, measured in centimetres from its equilibrium position, can be modelled by the second order differential equation:

$\ddot{x} + 5\dot{x} + 4x = 0$, where $t$ is the time in seconds after the initial displacement.

Given that $y = \dot{x}$, show that $\dot{y} = -4x - 5y$.

The differential equation can be expressed in the form $\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = A \begin{pmatrix} x \\ y \end{pmatrix}$, where A is a $2 \times 2$ matrix.

Write down the matrix A.

Find the eigenvalues of matrix A.

Find the eigenvectors of matrix A.

Given that at $t = 0$, the platform is displaced $6$ cm from equilibrium and its velocity is $3$ cm/s, find an expression for $x$ in terms of $t$.

A chemical compound is dissolving in a solvent. Initially, there are 225 grams of the compound. After 15 minutes, 144 grams of the compound remain undissolved.

The mass of the undissolved compound, M grams, remaining after t minutes, can be modelled by the differential equation $\frac{dM}{dt} = -k\sqrt{M}$, where k is a positive constant.

(a) Show that $M = (15 - \frac{t}{5})^2$.

(b) Calculate the time it takes for the entire compound to dissolve.

The concentration of a chemical, $C$ (in mol dm$^{-3}$), in a reaction vessel at time $t$ seconds is modelled by the differential equation

$\frac{\mathrm{d}^2C}{\mathrm{d}t^2} + 7\frac{\mathrm{d}C}{\mathrm{d}t} + 10C = 0$

(a) (i) Use the substitution $V = \frac{\mathrm{d}C}{\mathrm{d}t}$ to show that this equation can be written as

$\begin{pmatrix} \frac{\mathrm{d}C}{\mathrm{d}t} \\ \frac{\mathrm{d}V}{\mathrm{d}t} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -10 & -7 \end{pmatrix} \begin{pmatrix} C \\ V \end{pmatrix}$.

(ii) Find the eigenvalues for the matrix $\begin{pmatrix} 0 & 1 \\ -10 & -7 \end{pmatrix}$.

(iii) Hence state the long-term rate of change of the chemical concentration.

The chemical reaction is now subjected to an external input, and the equation for the concentration is amended to

$\frac{\mathrm{d}^2C}{\mathrm{d}t^2} + 7\frac{\mathrm{d}C}{\mathrm{d}t} + 10C = 2t + 5$.

(b) (i) Use the substitution $V = \frac{\mathrm{d}C}{\mathrm{d}t}$ to write the differential equation as a system of coupled, first order differential equations. State the initial conditions given that, at $t = 0$, the concentration of chemical C is $1$ mol dm$^{-3}$ and its rate of change is $0$ mol dm$^{-3}$ s$^{-1}$.

(ii) Use Euler's method with a step length of $0.1$ to find the concentration of the chemical when $t = 1$ s. Give your answer to three significant figures.

(iii) Find the long-term rate of change of the chemical concentration.

A chemical reaction is monitored in a laboratory. The concentration of a specific compound, C, in milligrams per litre (mg/L), changes over time, t, in minutes.

The rate of change of the compound's concentration is modelled by

$\frac{\mathrm{d}C}{\mathrm{d}t} = -0.5t + 10, \quad t \ge 0$

At 8 minutes, the concentration of the compound is 70 mg/L.

Find an expression for C in terms of t.

The experiment continues for an extended period.

Describe how the compound's concentration changes if the time elapsed is between 25 minutes and 35 minutes. Justify your answer.

A chemical engineer is studying the concentration of an intermediate product, $C$, in a reaction vessel. The change in concentration over time $t$ (in minutes) is modelled by the second order differential equation:

$\frac{d^2C}{dt^2} + 4\frac{dC}{dt} + 3C = 0$

where $t \ge 0$. It is known that when $t = 0$, the initial concentration is $C = 0$ and the rate of change of concentration is $\frac{dC}{dt} = 2$.

Show that the system of coupled first order equations:

$\frac{dC}{dt} = y$

$\frac{dy}{dt} = -3C - 4y$

can be written as the given second order differential equation.

Find the eigenvalues of the system of coupled first order equations given in part (a).

Hence find the exact solution of the second order differential equation, given the initial conditions $C(0) = 0$ and $\frac{dC}{dt}(0) = 2$.

Sketch the graph of $C$ against $t$ for $t \ge 0$, labelling the maximum point of the graph with its coordinates.

If the concentration of the intermediate product $C$ exceeds $0.2$ arbitrary units, the reaction needs to be monitored closely. Use the model to calculate the total amount of time (in minutes) during which the reaction needs to be monitored closely.

The chemical engineer decides to monitor the reaction for 15% longer than the time found from the model in part (e).

Write down one reason, with reference to the context, to support this decision.

The rate of change of the volume of water in a reservoir, in thousands of cubic meters per year, is given by the equation:

$\frac{\mathrm{d}V}{\mathrm{d}t} = 6t^2 - 20t$

where $V$ is the volume of water in thousands of cubic meters and $t$ is the time in years since the start of monitoring.

(a) Determine whether the volume of water in the reservoir is increasing or decreasing when $t=2$ years.

(b) One year after the start of monitoring, the volume of water in the reservoir was 5 thousand cubic meters. Find an expression for $V(t)$, the volume of water in the reservoir at time $t$, for $t \ge 0$.

A marine engineer is modelling the vertical displacement, $h$ meters, of a buoy from its equilibrium position at time $t$ minutes after being disturbed by a wave. The motion is described by the differential equation:

$\frac{\text{d}^2 h}{\text{d}t^2} + 6\frac{\text{d}h}{\text{d}t} + 8h = 0$.

This equation can be rewritten as a system of coupled first-order differential equations:

$\frac{\text{d}h}{\text{d}t} = y$

$\frac{\text{d}y}{\text{d}t} = -8h - 6y$.

Find the general solution for $h$.

Initially, the buoy is at its equilibrium position ($h = 0$) and has an initial upward velocity of $2$ m/min ($\frac{\text{d}h}{\text{d}t} = 2$).

Find an expression for $h$ in terms of $t$.

Sketch $h$ against $t$ in the interval $0 \le t \le 4$.

The engineer is interested in the maximum upward displacement of the buoy.

Find the time, in minutes, when this maximum displacement occurs.

Calculate the maximum upward displacement of the buoy.

An improved model for the buoy's motion, considering an external periodic force, is given by:

$\frac{\text{d}^2 h}{\text{d}t^2} + 6\frac{\text{d}h}{\text{d}t} + 8h = h \cos t$.

The same initial conditions as in part (b.i) apply ($h(0) = 0$ and $\frac{\text{d}h}{\text{d}t}(0) = 2$).

Use Euler's method with a $t$-interval of $0.1$ to predict the value of $h$ when $t = 1$.

A colony of bacteria is growing in a nutrient solution. The rate of change of the population, $P$, with respect to time, $t$ (in hours), is modelled by the differential equation $\frac{dP}{dt} = P \cos t (e^{-\sin t})$. At time $t = 0$, the population is $P = 10$.

(a) By using Euler's method with a step length of 0.1, find an approximate value for the population when $t = 0.3$. Give your answer to three significant figures.

(b) By solving the differential equation, find the percentage error in your approximation for the population when $t = 0.3$. Give your answer to three significant figures.

This question explores models for the temperature of a cooling metal object.

A metal object is heated and then allowed to cool in a room where the ambient temperature is $20$ °C. The temperature, $T$ °C, of the object is recorded every $5$ minutes, starting from $t = 0$ minutes.

The data is first modelled using a linear function, $T(t) = at + b$, where $a, b \in \mathbb{R}$.

Find the equation of the regression line of $T$ on $t$.

Interpret the meaning of the parameter $a$ in the context of the model.

Suggest why using this linear regression equation to predict the time it will take for the object to cool to $20$ °C could be unreliable.

The data is then modelled using a quadratic function, $T(t) = pt^2 + qt + r$, where $p, q, r \in \mathbb{R}$.

Find the equation of the least squares quadratic regression curve.

Use this quadratic equation to predict the time it will take for the object to cool to $40$ °C.

Hence, write down a suitable domain for the function $T(t) = pt^2 + qt + r$ in the context of this cooling process.

A metal object with a constant heat capacity $C$ Joules/°C is cooling in a room. The rate of heat transfer from the object to its surroundings is given by $P = \alpha A (T - T_a)$, where $P$ is the rate of heat loss in Joules/minute, $\alpha$ is the heat transfer coefficient, $A$ is the surface area of the object, $T$ is the object's temperature, and $T_a$ is the ambient temperature. The rate of change of the object's internal energy is given by $\frac{dE}{dt} = C \frac{dT}{dt}$.

Given that the ambient temperature is $T_a = 20$ °C and assuming that all heat loss is due to this process, show that the differential equation for the temperature of the object can be written as $\frac{dT}{dt} = -K(T - 20)$, where $K$ is a positive constant.

By solving the differential equation $\frac{dT}{dt} = -K(T - 20)$, show that the general solution is given by $T = 20 + Ae^{-Kt}$, where $A \in \mathbb{R}$.

Use the general solution from part (d) and the initial condition $T(0) = 90$ °C, along with the data point $T(5) = 74.5$ °C, to find the values of $A$ and $K$. Hence, predict the time it takes for the object to cool to $25$ °C.

If the object was initially heated to a different temperature, $T_0'$ °C, such that it cools to $30$ °C in $30$ minutes, find this new initial temperature $T_0'$. Use the value of $K \approx 0.0500589$ from part (e).

Now, consider a scenario where the object is cooling, but also receives a small amount of heat from an external source that decreases over time. The temperature of the object, $T$ °C, is modelled by the differential equation $\frac{dT}{dt} = -0.05(T - 20) + 1.5 e^{-0.1t}$.

Given the initial temperature $T(0) = 90$ °C, use Euler's method with a step length of $h = 2$ minutes to estimate the temperature of the object at $t = 6$ minutes.

The rate of change of the concentration of a chemical compound, $C$ (in mol/L), in a reaction vessel with respect to time, $t$ (in minutes), is modeled by the differential equation

$(t^3+1)\frac{dC}{dt} = \frac{t^2}{2C-6}$, for $t \ge 0, C \ge 3$.

Given that $C = 3$ when $t = 0$.

Explain why Euler's method cannot be used to find an approximate value for $C$ when $t = 0.1$.

By solving the differential equation, show that $C = 3+\sqrt{\frac{1}{3}\ln(t^3+1)}$.

Hence deduce the value of $C$ when $t = 0.2$.

A chemical spill has contaminated a section of a river. Environmental engineers are monitoring the concentration of a particular pollutant. Let $P(t)$, measured in milligrams per litre (mgL$^{-1}$), be the concentration of the pollutant, $t$ days after a new batch of pollutant is introduced. The rate at which the pollutant naturally degrades or is flushed away is modelled as directly proportional to its concentration, leading to the differential equation

$\frac{dP}{dt} = -kP$, where $k \in \mathbb{R}^+$.

The initial concentration is $P_0$ mgL$^{-1}$, $P_0 > 0$.

By solving the differential equation, show that $P = P_0 e^{-kt}$.

For the remainder of this question, you will consider this pollutant where it is known that $k = 0.15$. The first significant spill occurs at time $t = 0$ and it is assumed that before this there is no pollutant present in the river.

Find the time, in days, for this pollutant to reach 10% of its initial concentration.

The pollutant is added to the river every $T$ days due to regular discharges, and in constant amounts, such that the concentration of the pollutant is increased by an amount $P_0$ mgL$^{-1}$. To simplify the model, it is assumed that each time the pollutant is added, the concentration in the river increases instantaneously.

Show that the concentration of the pollutant is $P_0(1+e^{-0.15T} + e^{-0.30T})$ immediately after the third discharge is given.

Immediately after the $n^{th}$ discharge is given, the concentration of the pollutant is

$P_0(1+e^{-0.15T} + e^{-0.30T} + ... + e^{-0.15(n-1)T})$.

Show that this concentration can be expressed as $P_0\left(\frac{1-e^{-0.15nT}}{1-e^{-0.15T}}\right)$.

After the river has been subjected to these discharges for a long time, it is required to keep the pollutant concentration within a particular range to ensure ecological safety.

Let $H_n$ be the highest concentration of the pollutant in the river for the interval $(n-1)T < t < nT$.

Let $L_n$ be the lowest concentration of the pollutant in the river for the interval $(n-1)T < t < nT$.

This is shown in the following graph.

$H_\infty$ is defined as $\lim_{n\to\infty} H_n$ and $L_\infty$ is defined as $\lim_{n\to\infty} L_n$.

Find, in terms of $P_0$ and $T$, an expression for

$H_\infty$.

$L_\infty$.

Show that

$H_\infty - L_\infty = P_0$.

$\frac{1}{0.15} \ln\left(\frac{H_\infty}{L_\infty}\right) = T$.

It is known that this pollutant is considered safe if the long-term concentration never exceeds $0.50 \text{ mgL}^{-1}$ and ecologically effective if it never drops below $0.10 \text{ mgL}^{-1}$.

Hence, for this pollutant, find a suitable value for

$P_0$.

$T$.

For the values of $P_0$ and $T$ found in part (g), find the proportion of time for which the concentration of the pollutant is at least $0.10 \text{ mgL}^{-1}$ between the first and second discharges.

Suggest a reason why the environmental regulations might specify a different value for $T$ to that found in part (g)(ii).

A chemical reaction's rate is modelled by the differential equation $\frac{dy}{dx} = \frac{2y}{x}$, where $y$ represents the concentration of a product at time $x$ (in minutes), for $x > 0$ and $y > 0$. At $x = 1$ minute, the concentration is $y = 3$ units.

(a) Use Euler's method with a step size of $0.5$ to find an approximation for the concentration at $x = 2$ minutes. For each value of $x$, show the corresponding value of $y$ in your working.

(b) Solve the differential equation $\frac{dy}{dx} = \frac{2y}{x}$, giving the answer in the form $y = f(x)$ for some function $f$.

(c) Hence, find the exact value of the concentration at $x = 2$ minutes.

(d) Find the absolute percentage error in the value of $y(2)$ given by Euler's method. Give your answer to $2$ significant figures.

A small scientific probe is launched into a dense liquid to collect data. Its descent is affected by gravity, buoyancy, and liquid resistance. The direction downwards is taken to be positive.

Initially, a simple model for the probe's velocity, $v$ ms$^{-1}$, at time $t$ seconds, assumes constant effective acceleration due to gravity and buoyancy, $g'$ ms$^{-2}$, given by:

$\frac{dv}{dt} = g'$

When the probe enters the liquid at $s = 0$ m, its initial velocity is $v = 5$ ms$^{-1}$. The displacement from its initial position is $s$ metres.

(i) Use the chain rule to show that $\frac{dv}{dt} = v\frac{dv}{ds}$.

(ii) Assuming that $g'$ is a constant, solve the differential equation $v\frac{dv}{ds} = g'$ to find $v$ as a function of $s$.

(iii) Using $g' = 9.8$ ms$^{-2}$, determine whether the model predicts that the probe will reach a velocity of $15$ ms$^{-1}$ at some point before it reaches a depth of $s = 200$ m. Justify your answer.

To test the model $\frac{dv}{dt}=g'$, the probe conducted a trial descent, and data for $v$ against $t$ was recorded.

(i) If the model is correct, describe the shape of the graph of $v$ against $t$.

(ii) The observed data showed a graph where the velocity increased rapidly at first, then its rate of increase slowed down, eventually approaching a constant value. Use this observation to comment on the validity of the model in part (a).

An improved model considers liquid resistance, using

$\frac{dv}{dt} = g'-k'v^2$

where $k'$ is a positive constant. You are reminded that initially $s = 0$ and $v = 5$. You may assume that $g' - k'v^2 > 0$.

(i) By using $\frac{dv}{dt} = v\frac{dv}{ds}$, solve the differential equation to find $v$ in terms of $s$, $g'$ and $k'$.

The probe's engineers use the graph of $v$ against $t$ from the trial descent to estimate the value of $k'$.

(ii) The gradient $\frac{dv}{dt}$ is estimated to be $3.05$ ms$^{-2}$ when $v = 15$ ms$^{-1}$. Taking $g'$ to be $9.8$ ms$^{-2}$, use this information to show that the engineers found that $k' = 0.03$.

(iii) Hence, find the value of $v$ predicted by this model, as $s$ tends to infinity.

(iv) Find the upper bound for the velocity according to this model, given that $0 < s \le 200$. Give your answer to four significant figures.

(d) A more refined model for the probe's descent suggests that the rate of change of velocity with respect to displacement is given by $\frac{dv}{ds} = \frac{5000}{(1000-s)^2} - 0.0001 v^2$. Use Euler's method with a step length of $50$ m to estimate the value of $v$ when $s = 200$ m. Take the initial velocity $v = 5$ ms$^{-1}$ at $s = 0$ m.

(i) Suggest one improvement to the use of Euler's method which might increase the accuracy of the prediction of the model.

(ii) Suggest one factor not explicitly considered by the model in part (d) which might lead to a difference between the model's prediction and the data collected.