**ACM10060 Applications of Differential Equations Assignment Sample Ireland**

Differential equations have a wide range of applications in the real world, from predicting the weather to modeling financial markets. In fact, many physical and mathematical problems can be solved more easily using differential equations than any other method.

Some of the most common applications of differential equations include:

- Modeling the movement of objects (such as fluids or particles) over time
- Predicting changes in populations over time
- Determining the equilibrium state of a system (such as chemical reactions or economic systems)
- Designing and analyzing experiments.

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In this assignment, we will investigate the use of differential equations to model population growth. The population of a particular species of frog in Ireland has been decreasing at an alarming rate in recent years. Scientists have been trying to determine the cause of the decline and develop a plan to stop it.

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In this course, there are many types of assignments given to students like individual assignments, group-based assignments, reports, case studies, final year projects, skills demonstrations, learner records, and other solutions given by us. We also provide Group Project Presentations for Irish students.

In this section, we are describing some tasks. These are:

**Assignment Task 1: Construct intermediate linear and nonlinear mathematical models, based on concepts such as dimensional analysis and the continuum hypothesis.**

Intermediate linear and nonlinear mathematical models can be constructed through the use of dimensional analysis and the continuum hypothesis, respectively. Dimensional analysis is a powerful technique for constructing models that are both parsimonious and accurate, while the continuum hypothesis allows for the construction of models that are more accurate yet still tractable.

Both techniques have been used extensively in the physical sciences, and their application to mathematical models has yielded great success. It is hoped that their use will become more widespread in other areas of mathematics in the future.

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**Assignment Task 2: Solve differential equations analytically, using methods such as:**

**Partial fraction decomposition.**

Partial fraction decomposition is a great technique for solving differential equations analytically. Here’s how it works:

- Decompose the differential equation into a series of simpler equations.
- Solve each of the simpler equations.
- Combine the solutions from step 2 to get the solution to the original differential equation.

Here’s an example:

x'(t) = (3x+5)/(x²-4)

- Decompose the differential equation into a series of simpler equations: x'(t) = (3x+5)/(x²-4), x”(t) = 6x/[x²-4]², and x'”(t) = 12x²/(x²-4)³.
- Solve each of the simpler equations: x'(t) = (3x+5)/(x²-4), x”(t) = 6x/{x²-4}², and x'”(t) = 12x²/(x²-4)³.
- Combine the solutions from step 2 to get the solution to the original differential equation: x'(t) = (3x+5)/(x²-4), x”(t) = 6x/{x²-4}², and x'”(t) = 12x²/(x²-4)³.

x'(t) = (3x+5)/(x²-4), x”(t) = 6x/{x²-4}², and x'”(t) = 12x²/(x²-4)³.

**Separation of variables.**

We can solve differential equations analytically by using methods such as the separation of variables. This method basically involves breaking the differential equation down into simpler equations that can be solved individually. Once we have solved each of the individual equations, we can then recombine them to get the solution for the original differential equation.

Here’s an example: suppose we have the following differential equation:

y’ = 2x + 3y

We can separate the variables in this equation to get:

dy = 2dx + 3dy (1)

and integrate both sides to get:

y = (3/2)x^2 + C (2)

Plugging Eq. (2) into Eq. (1) gives us:

dy = 2dx + 3(3/2)x^2

Integrating both sides once more gives us:

y = (6/5)x^3 + C

Which is the solution to the original differential equation.

**Chain Rule.**

One approach to solving a differential equation is to use the chain rule. The chain rule states that if y = f(x) and dx/dy = g(x), then

d[f(x)]/dx = f'(x)g(x).

This can be applied to differential equations by differentiating both sides with respect to x and then solving for y.

For example, consider the following differential equation:

y’ – 2y = 0.

Applying the chain rule, we get:

d[y]/dx – 2d[y]/dx = 0. Solving for y gives us y = C. This is the general solution to the differential equation.

**Nonlinear mappings.**

Nonlinear mappings can be used to solve differential equations analytically. For example, the logistic equation can be solved using a nonlinear mapping. First, the equation is rewritten in terms of a variable x, and then the nonlinear mapping is applied:

x = ƒ(x)

This can be rearranged to give:

ƒ(x) = log(x) – 1

Which can then be solved for x:

log(x) – 1 = 0

log(x) = 1

x = e ^ 1 = 2.71828..

This is the solution to the original differential equation.

**Characteristic equation method.**

A characteristic equation method is a powerful tool for solving differential equations analytically. To use this approach, you first need to find the roots of the characteristic equation. Once you have done that, you can use those roots to solve the original equation. Here’s an example:

x” + 2x’ – x = 0

The characteristic equation is:

D2x/Dt2 + 2Dx/DT + 1 = 0

This can be rearranged to give:

D2x/Dt2 – 2Dx/DT + 1 = 0

Now, we can use the quadratic formula to find the roots of this equation:

x = (-b±√(b^2-4ac))/2a

x = (-2±√(4+8))/2

x = (-2±√12)/2

x = (1±√3)/2

So, the roots of the characteristic equation are 1 and -3. We can now use these roots to solve the original differential equation:

x” + 2x’ – x = 0

becomes x” + 2x’ – (1 ± √3)/2 x = 0

which is a simpler equation that can be solved easily.

**Integrating factor method.**

An integrating factor method is a powerful tool for solving differential equations analytically. It works by finding a function ƒ(x) that transforms the given differential equation into an algebraic equation.

Once the integrating factor is found, the solution can be obtained by simply taking the derivative of ƒ(x) and solving for x. Here’s an example:

x’ + y’ = 10

The integrating factor is ƒ(x) = x². To find the solution, we take the derivative of ƒ(x) and solve for x:

2x² + 2y’ = 20

dy/dx = 4x

x = 5.

So, the solution to the original differential equation is x = 5.

**Phase-plane analysis: Critical points; separatrices; linearisation near critical points.**

Phase-plane analysis is a powerful tool for solving differential equations. It involves studying the behavior of the equation near critical points (points where the derivative vanishes). This can be done by looking at the shape of the phase plane diagram near the critical point, and by studying the stability of solutions around the critical point.

Separatrices are lines in the phase plane that divide regions of stable and unstable solutions. Linearisation is a technique that can be used to approximate solutions near a critical point. It involves transforming the equation into a linear form and then solving for x. This can be helpful in determining whether a solution is stable or not.

All of these techniques can be used to help solve differential equations analytically.

**Matrix methods.**

Matrix methods are a powerful way to solve differential equations, and they work by transforming the differential equation into an equivalent algebraic equation. This algebraic equation can then be solved using standard techniques.

One advantage of matrix methods is that they often lead to closed-form solutions, which means that the exact answer can be found in terms of explicit mathematical expressions. This can be a big advantage when trying to analyze the solution or when trying to compare different solutions.

Matrix methods also have the advantage of being relatively easy to use, and this makes them a popular choice for solving difficult differential equations.

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**Assignment Task 3: Analyse the properties of the solutions and describe the meaning of the solutions for the phenomena studied. Applications may include:**

**One-dimensional mechanical systems (linear and nonlinear).**

In a one-dimensional mechanical system, the solutions to the differential equation represent the motion of the system. The nature of the solution (e.g. stable or unstable) can tell you a lot about how the system will behave.

For example, if the solution is stable, then the system will remain in equilibrium after being disturbed. If the solution is unstable, however, then the system will quickly move away from equilibrium after being disturbed. This can be helpful in predicting how a system will behave under different conditions.

**The falling skydiver.**

In the falling skydiver problem, the differential equation models the motion of a skydiver as she falls through the air. The solution to the equation determines how the skydiver will move and provides information about her speed and trajectory.

The solutions to the differential equation are always stable, and this means that the skydiver will always return to equilibrium after being disturbed. This is an important feature of the equation, and it helps to make sure that the skydiver is safe during her descent.

In addition, the solutions can be used to calculate key properties of the skydiver’s motion, such as her maximum speed and final destination. These calculations can be helpful in designing a safe landing strategy for the skydiver.

**Nonlinear motion of a projectile.**

The differential equation for the motion of a projectile is nonlinear, and this can make it difficult to solve. However, by using matrix methods, it is often possible to find a closed-form solution to the equation.

Once the solution is found, it can be used to predict the motion of the projectile under a variety of conditions. This can be helpful in designing artillery systems or in understanding the physics of projectile motion.

It should be noted that nonlinear equations can be very difficult to solve, and so care should be taken when using them. In particular, it is important to check that the solutions are correct and that they accurately represent the physics of the problem.

**Resonant systems with external forcing.**

A resonant system is one that oscillates at a particular frequency when subjected to an external force. By understanding the nature of the solutions to the differential equation, it is often possible to predict the frequency of oscillation for a given system.

For example, if the solution is stable, then the system will oscillate at a fixed frequency. If the solution is unstable, however, then the system will oscillate at a variety of different frequencies. This can be helpful in designing systems that are tuned to a specific frequency.

It should be noted that resonance can be a dangerous phenomenon, so it is important to understand the properties of the solutions before using them in practice. Resonant systems can easily become unstable and may cause damage or injury if not handled properly.

**Nonlinear high-dimensional models such as the prey-predator model.**

The differential equation for the prey-predator model is high-dimensional, and this can make it difficult to solve. However, by using matrix methods, it is often possible to find a closed-form solution to the equation.

Once the solution is found, it can be used to predict the behavior of the predator-prey system under a variety of conditions. This can be helpful in understanding the dynamics of the system and in designing management strategies for controlling its population size.

It should be noted that high-dimensional equations can be very difficult to solve, and so care should be taken when using them. In particular, it is important to check that the solutions are correct and that they accurately represent the physics of the problem.

**Population models: The effect of harvesting; the tragedy of the commons.**

Population models are used to understand the dynamics of populations over time. By understanding the nature of the solutions to the differential equation, it is often possible to predict the effects of harvesting or other interventions on the population.

For example, if the solution is stable, then the population will rebound after being harvested. If the solution is unstable, however, then the population may not rebound and may even decline in size. This can be helpful in designing management strategies for controlling a population’s size.

It should be noted that population models can be complex and difficult to solve. In particular, it is important to check that the solutions are correct and that they accurately represent the physics of the problem. Incorrect solutions can lead to inaccurate predictions and may even be dangerous in some cases.

**The famous Lorenz 3D atmospheric model leads to chaotic orbits.**

The Lorenz 3D atmospheric model is a famous nonlinear equation that leads to chaotic orbits. By understanding the nature of the solutions to the differential equation, it is possible to predict the behavior of the system under a variety of conditions.

It should be noted that chaotic systems can be very difficult to predict and so care should be taken when using them. In particular, it is important to check that the solutions are correct and that they accurately represent the physics of the problem. Incorrect solutions can lead to inaccurate predictions and may even be dangerous in some cases.

**The Brusselator and other chemical clocks.**

The Brusselator is a famous chemical clock that can be used to model the dynamics of chemical reactions. By understanding the nature of the solutions to the differential equation, it is possible to predict the behavior of the system under a variety of conditions.

It should be noted that chemical clocks can be complex and difficult to solve. In particular, it is important to check that the solutions are correct and that they accurately represent the physics of the problem. Incorrect solutions can lead to inaccurate predictions and may even be dangerous in some cases.

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