Euler method for First order differential equations (single and coupled systems): notes and practice questions

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 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.

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 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 $t = 1$.

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 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.

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.

In this question, marine biologists are studying the population dynamics of a rare species of "Azurefin" fish in a newly established protected reef area.

Historically, the Azurefin population in this region maintained a stable size of $5000$ individuals. Following a period of environmental disturbance, the population was reduced to $1000$ fish. At this point, the area was designated a protected marine reserve, and conservation efforts began, leading to a recovery in the fish population.

Researchers wish to model the size of the Azurefin population, $x$, as a function of $t$, where $t$ is the time, in years, since the establishment of the protected reserve.

Initially, the researchers consider using the logistic model:

$x = \frac{L}{1+Ce^{-kt}}$, where $L, C, k \in \mathbb{R}^+$.

The researchers decide to set $L = 5000$.

State the assumption being made by setting $L = 5000$.

At $t = 0$, the population of Azurefin fish is $1000$.

Find the value of $C$.

At $t = 3$ years, the population of Azurefin fish is found to have increased to $2500$.

Find the value of $k$. Give your answer correct to three significant figures.

Use your model to predict the size of the Azurefin population in the area $6$ years after it became protected. Give your answer correct to the nearest whole number.

An alternative model for population growth is called the Gompertz model. When applied by the researchers to the Azurefin population, this model satisfies the differential equation:

$\frac{dx}{dt} = ax \ln \left( \frac{5000}{x} \right)$, $a \in \mathbb{R}^+$.

Write down the value of $\frac{dx}{dt}$ when $x = 5000$.

Interpret your answer to part (b)(i) in context.

Consider the function $f(x) = \ln (\ln 5000 - \ln x)$, where $0 < x < 5000$.

Show that $f'(x) = \frac{-1}{x \ln \left( \frac{5000}{x} \right)}$.

Hence, use separation of variables to show that the general solution of

$\frac{dx}{dt} = ax \ln \left( \frac{5000}{x} \right)$, where $0 < x < 5000$,

can be written as

$\ln x = \ln 5000 - Ae^{-at}$,

where $A$ is an arbitrary positive constant.

Use the size of the Azurefin population at $t = 0$ to find the value of $A$.

Give your answer in the form $A = \ln p$, where $p \in \mathbb{Z}^+$.

Use the size of the Azurefin population at $t = 3$, given in part (a), to show that $a = 0.281$, correct to three significant figures.

Use the Gompertz model to predict the size of the Azurefin population at $t = 6$. Give your answer correct to the nearest whole number.

After $6$ years, the Azurefin population is measured and is found to be $3200$.

Comment on the predictions made by the two models.

By tracking individual Azurefin fish, the researchers find that about $5\%$ of the population migrates out of the protected area each year.

They decide to adapt the Gompertz model to allow for this. The new model will satisfy the differential equation:

$\frac{dx}{dt} = 0.280799x \ln \left( \frac{5000}{x} \right) - 0.05x$.

Use Euler's method, with a step size of $0.5$ years and an initial value of $x_0 = 2500$ when $t = 3$, to find an estimate for the size of the Azurefin population when $t = 6$.

Give your answer correct to the nearest whole number.

Comment on your answer.

A chemical engineer is studying the concentration of a pollutant, $P$, in mol/L, in a closed system. The rate of change of the pollutant's concentration at time $t$, in hours, is modelled by the differential equation

$10 = 1.5\frac{d^2P}{dt^2} + 0.8\frac{dP}{dt} + 5P$.

At the start of the experiment ($t=0$), the concentration of the pollutant is $2.0$ mol/L and its rate of change is $-3.0$ mol/L per hour.

By using Euler's method with a step size of $h = 0.05$ hours, find the estimated concentration of the pollutant after $0.2$ hours.

Find the first estimated time the concentration of the pollutant reaches $1.0$ mol/L.

After $0.2$ hours, the chemical engineer measures the concentration of the pollutant as $1.65$ mol/L, accurate to the nearest $0.01$ mol/L.

Calculate the maximum possible percentage error between your estimate from part (a)(i) and the measured concentration after $0.2$ hours.

A biologist is studying the interaction between two species of microorganisms, X and Y, in a controlled environment. The population dynamics are modelled by the system of coupled differential equations:

$\frac{dx}{dt} = x + 2y$

$\frac{dy}{dt} = -2x + y$

The following phase portrait illustrates the general behaviour of trajectories for this system.

Write down which one of the following could be an eigenvalue for the system.

A. $1$

B. $2i$

C. $1 + 2i$

D. $-1 + 2i$

Using Euler's method with a step length of $0.05$, find the approximate values of $x$ and $y$ when $t = 0.20$, given that at $t = 0$, $x = 2$ and $y = 1$. Give your answers to three significant figures.