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# ACM10060 Applications of Differential Equations UCD Assignment Example

Differential equations are equations that relate a function with one or more of its derivatives. Differential equations are useful for modeling many real-world situations, such as population growth, the spread of diseases, and the motion of objects. In physics, differential equations are used to describe the behavior of systems under a variety of conditions. In engineering, they can be used to design and optimize complex systems. In mathematics, differential equations can be used to solve difficult problems that cannot be solved using traditional methods.

There are many different types of differential equations, but they can all be classified into one of four categories: linear, separable, exact, or integrating. Linear differential equations are the simplest type and can be solved using a variety of methods. Separable differential equations can be solved by splitting the equation into two parts and solving each part separately. Exact differential equations can be solved using a variety of methods, but usually require some algebraic manipulation. Integrating differential equations can be done by hand, but is usually done using a computer.

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

There are several ways to construct intermediate linear and nonlinear mathematical models. One approach is dimensional analysis, which can be used to develop simplified models that capture the essential features of a system without being excessively complex. Another approach is the continuum hypothesis, which states that certain physical properties (such as mass) are continuous rather than discrete. This can be used to develop more accurate models, although it may also make them more difficult to work with. Ultimately, the choice of approach depends on the specific needs of the application.

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

### Partial fraction decomposition

Partial fraction decomposition is a great way to solve differential equations analytically. It works by separating the differential equation into simpler terms, which can then be easily solved.

To use partial fraction decomposition to solve a differential equation, you’ll first need to identify all of the partial fractions in the equation. Next, you’ll need to find the constants of integration for each partial fraction. Finally, you can solve each term using standard techniques.

It’s important to note that partial fraction decomposition will only work if the original differential equation can be reduced to a linear form. If it can’t be reduced to a linear form, then you’ll need to use other methods (such as Laplace transforms) to solve it.

### Separation of variables

The method of separation of variables is a powerful technique for solving differential equations. It works by splitting the equation into two parts, one involving only derivatives with respect to x, and the other involving only derivatives with respect to y. We then solve each part separately, using the appropriate techniques.

For example, consider the following equation:

We can split this into two parts: one involving derivatives with respect to x, and the other involving derivatives with respect to y. The first part becomes:

And the second part becomes:

We can now solve each part separately. The first part can be solved using integration by parts, while the second part can be solved using a standard technique such as substitution or elimination.

### Chain rule

If you want to solve differential equations analytically using the chain rule, you will need to first understand what the chain rule is and how it works. The chain rule states that if you have a function that is a composition of two other functions, then the derivative of that function can be calculated by taking the derivative of each of the individual functions and multiplying them together.

For example, let’s say you have a function f(x) = g(h(x)). If we take the derivative of this function using the chain rule, we would get:

df/dx = dg/dh * dh/dx

In other words, we take the derivative of each function (g and h) and multiply them together.

This is a powerful tool for solving differential equations, as it allows us to break the problem down into simpler parts. By taking the derivative of each function and multiplying them together, we can often obtain a simpler equation that can be solved more easily.

### Nonlinear mappings

If you want to solve a differential equation using a nonlinear mapping, you’ll need to first understand what a nonlinear mapping is and how it works. A nonlinear mapping is a function that maps one or more variables in a complex way. This can make the solution to the differential equation very difficult (or even impossible) to obtain.

However, some techniques can be used to simplify the problem and make it easier to solve. For example, you can use Newton’s Method or the bisection method to find approximate solutions to the differential equation. Alternatively, you can use WolframAlpha or another online tool to help you visualize the solution.

### Characteristic equation method

The characteristic equation method is a powerful technique for solving differential equations. It works by solving a special equation that is related to the differential equation. This equation can often be solved relatively easily, which then allows us to solve the differential equation itself.

The characteristic equation method is based on the following theorem:

If F(x, y) is a function that satisfies the differential equation dy/dx = F(x, y), then there exists a function X(x) such that F(x, X(x)) = 0.

In other words, if we can find a function X(x) that satisfies the condition F(x, X(x)) = 0, then we can use this function to solve the original differential equation.

### Integrating factor method

The integrating factor method is a numerical technique used to solve ordinary differential equations. It’s a very powerful method and can be used to solve a variety of problems, including problems with discontinuities or multiple solutions.

The basic idea behind the method is to transform the original ODE into an algebraic equation by introducing an integrating factor. This integrating factor can then be integrated to give a solution to the original ODE. There are several different methods for finding the integrating factor, but one of the most common is to use the eigenvalues and eigenvectors of the matrix associated with the ODE.

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

Phase-plane analysis is a powerful tool for solving dynamic systems. It allows us to visualize the solutions to the system and determine where they are stable or unstable.

The basic idea behind phase-plane analysis is to graph the solutions to the system in two dimensions. This can be done by plotting the x- and y-values of the solutions as they change over time. By studying the graph, we can determine where the solutions are stable or unstable and identify any critical points of separatrices.

We can also use linearisation near critical points to get a better understanding of how the system behaves near those points. This technique allows us to linearise the system around a particular point and then study the behavior of the linearized system. This can be helpful for determining whether a solution is stable or not.

### Matrix methods

Matrix methods are a powerful family of techniques for solving differential equations. They allow us to solve problems that would be impossible to solve using other methods.

The basic idea behind matrix methods is to break the differential equation down into a series of smaller equations. These smaller equations can then be solved using methods like the characteristic equation method or the integrating factor method. By breaking the problem down into smaller pieces, we can make it much easier to solve.

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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)

There are various types of one-dimensional mechanical systems, including linear and nonlinear systems. Each type has its own advantages and disadvantages, so it is important to choose the right system for the task at hand. Linear systems are typically easier to analyze and control, but they can be less effective in certain situations. Nonlinear systems may be more difficult to work with, but they can offer greater flexibility and power. Ultimately, it is up to the engineer to determine which type of system is best suited for the task at hand.

One-dimensional systems can be used to model a variety of different phenomena, including linear and nonlinear systems. Linear systems are often used to model simple mechanical problems, while nonlinear systems can be used to model more complex problems. In either case, it is important to understand the properties of the solutions to accurately predict the behavior of the system.

### The falling skydiver

The falling skydiver is a phenomenon studied in physics. In this scenario, a skydiver jumps from a high altitude and free falls for a certain amount of time before pulling the parachute chord to slow his descent.

The solutions to this problem can be found by using equations of motion along with the principles of aerodynamics. When solving for the position and velocity of the skydiver at any point during his descent, one must take into account the forces of gravity, air resistance, and drag (the force exerted by air molecules on an object).

The solutions to this problem are important because they help us better understand how skydivers safely descend from high altitudes. They also provide insights into the aerodynamic properties of falling objects.

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### Nonlinear motion of a projectile

The solutions to the nonlinear motion of a projectile can be classified into three categories: underdamped, overdamped, and critically damped. Each category has its own distinct set of properties that affect the motion of the projectile.

Underdamped solutions have a slowly decaying oscillatory behavior. This means that the projectile will continue to oscillate around the point of impact even after it has stopped moving. Overdamped solutions on the other hand exhibit a fast decaying behavior, meaning that the projectile will quickly come to rest after impact. Critically damped solutions lie in between these two extremes, with a decay rate that is just enough to avoid oscillations.

### Resonant systems with external forcing

Resonant systems with external forcing are systems that are driven by an external force at a specific frequency. When the external force is applied, the system will oscillate at the same frequency as the driving force.

The solutions to this problem are important because they can be used to produce oscillations in a system. This can be useful for a variety of purposes, such as generating power or testing the response of a system to a particular frequency. It can also be used to find the natural frequency of a system, which is the frequency at which the system will oscillate without any external input.

### Nonlinear high-dimensional models such as the prey-predator model

Nonlinear high-dimensional models are used to model complex systems with a large number of variables. These models can be used to predict the behavior of the system over time, and they can also be used to find solutions for specific problems.

The prey-predator model is a good example of a nonlinear high-dimensional model. This model can be used to study the dynamics of predator-prey interactions. It can also be used to predict the population size of both the predator and prey populations over time. This information is important for managing resources and preventing population crashes.

### Population models: The effect of harvesting; the tragedy of the commons

Population models are used to study the dynamics of population growth. They can be used to predict the impact of various factors on population size, including birth rates, death rates, and harvesting.

The effect of harvesting is an important topic in population modeling. When a population is harvested, its size is reduced. This has a ripple effect on the rest of the population, as it can lead to a decrease in the birth rate and an increase in the death rate.

The tragedy of the commons is another important topic in population modeling. This is the phenomenon where a shared resource is used to such an extent that it becomes depleted. This can happen with natural resources like fish or water, or it can happen with man-made resources like cars or parking spaces.

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### The famous Lorenz 3D atmospheric model leads to chaotic orbits

The Lorenz 3D atmospheric model is a well-known example of a chaotic system. This model can be used to study the dynamics of the atmosphere. It can also be used to predict the weather over time.

The Lorenz 3D atmospheric model is famous for its chaotic orbits. These orbits are caused by the sensitivity of the system to initial conditions. This means that even a small change in the starting point can lead to a completely different outcome. This makes it difficult to predict the weather using this model.

### The Brusselator and other chemical clocks

The Brusselator is a well-known example of a chemical clock. This is a system that can be used to study the dynamics of chemical reactions. It can also be used to predict the progress of a reaction over time.

The Brusselator is famous for its ability to produce sustained oscillations. These oscillations are caused by the positive feedback that is present in the system. This feedback leads to the amplification of small perturbations, which causes the reaction to oscillate over time.

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