# Modelling with Differential Equations

#### Prerequisite Knowledge

1. Complex Numbers:
• Basic operations (addition, subtraction, multiplication, division) with complex numbers.
• Polar and trigonometric forms.
• Use and application of De Moivre’s theorem.
• Argand diagrams and geometric interpretation of complex numbers.
• Complex conjugates and modulus-argument form.
2. Methods in Differential Equations:
• Formation of differential equations to represent given scenarios.
• First and second-order differential equations solutions.
• Familiarity with homogeneous and non-homogeneous equations.
• Application of methods like the separation of variables and integrating factor.
• Understanding of boundary and initial conditions.
3. Integration:
• Fundamental integration techniques: substitution, integration by parts, partial fractions.
• Definite and indefinite integrals.
• Integration involving trigonometric, exponential, and logarithmic functions.
• Use of integration to find areas under curves and volumes of revolution.
• Familiarity with improper integrals.
4. Differentiation:
• Chain, product, and quotient rules for differentiation.
• Differentiation of implicit functions.
• Higher-order derivatives.
• Differentiation of parametric and polar functions.
• Applications of differentiation: maxima, minima, points of inflection, and rate of change problems.

#### Success Criteria

Model Real-life Situations including Motion and Mixtures with First-order Differential Equations

• Successfully model real-world scenarios involving motion, like a falling object or a car’s acceleration, using first-order differential equations.
• Develop differential equations that describe mixing scenarios, such as the rate at which two liquids combine in a tank.
• Solve the derived equations, determining variables like time to reach a certain speed or concentration level in a mixture.
• Critically evaluate the appropriateness and accuracy of a given model in describing a real-life situation.

Use Differential Equations to Model Simple Harmonic Motion (SHM)

• Translate situations involving pendulums, springs, or other SHM examples into their respective differential equations.
• Recognise and describe the features of SHM, including equilibrium position, amplitude, and periodicity.
• Solve the differential equations to ascertain quantities like maximum displacement or period of oscillation.
• Relate mathematical solutions to practical scenarios, such as predicting the behaviour of a pendulum over time.

Model Damped and Forced Oscillations Using Differential Equations

• Represent real-world situations of damped oscillations, such as a pendulum in a viscous fluid, using differential equations.
• Derive equations for forced oscillations based on scenarios like a child being pushed on a swing.
• Solve these equations to determine characteristics like damping factor or frequency of forced oscillation.
• Apply the solutions to predict behaviours, like how long it takes for a pendulum’s amplitude to halve due to damping.

Model Real-life Situations using Coupled First-order Differential Equations

• Formulate systems of equations for situations where two entities are interdependent, such as predator-prey models.
• Solve the coupled differential equations, determining quantities like population sizes over time or interrelated velocities.
• Interpret and apply the mathematical results to predict trends or behaviours in the original real-world situation.
• Assess the validity of the coupled model in capturing the intricacies of the scenario it represents.

#### Teaching Points

• Model real-life situations including motion and mixtures with first-order differential equations:
• Differential equations can be used to model real-world situations that involve change over time.
• First-order differential equations are the simplest type of differential equations.
• There are a number of methods for solving first-order differential equations, such as separation of variables, integrating factors, and numerical methods.
• The solution of a first-order differential equation can be used to predict the future behaviour of the real-world situation it models.
• Examples of real-world situations that first-order differential equations can model include the motion of a falling object, the mixing of two fluids, and the growth of a population.
• Use differential equations to model simple harmonic motion:
• Simple harmonic motion is a type of periodic motion.
• The differential equation that models simple harmonic motion is called the equation of motion.
• The solution of the equation of motion can be used to predict the position, velocity, and acceleration of the object undergoing simple harmonic motion.
• Examples of real-world systems that exhibit simple harmonic motion include pendulums and springs.
• Model damped and forced oscillations using differential equations:
• Damped and forced oscillations are types of periodic motion that are affected by friction or external forces.
• The differential equations that model damped and forced oscillations are more complex than the equation of motion for simple harmonic motion.
• The solution of the differential equation can be used to predict the position, velocity, and acceleration of the object undergoing damped or forced oscillations.
• Examples of real-world systems that exhibit damped and forced oscillations include vibrating strings and electrical circuits.
• Model real-life situations using coupled first-order differential equations:
• Coupled first-order differential equations are a system of two or more differential equations that are related to each other.
• Coupled first-order differential equations can be used to model real-world situations that involve multiple interacting components.
• The solution of a coupled system of differential equations can be used to predict the future behaviour of the real-world situation it models.
• Examples of real-world situations that can be modelled by coupled first-order differential equations include the spread of disease and the interaction of predator and prey populations.

#### Common Misconceptions

• Model real-life situations, including motion and mixtures with first-order differential equations:
• Misunderstanding the difference between a differential equation and a mathematical model.
• Not being able to identify the correct differential equation to model a given real-world situation.
• Making mistakes when solving first-order differential equations.
• Not being able to interpret the solution of a first-order differential equation in the context of the real-world situation it models.
• Use differential equations to model simple harmonic motion:
• Misunderstanding the concept of simple harmonic motion.
• Not being able to identify the correct differential equation that models simple harmonic motion.
• Making mistakes when solving the differential equation for simple harmonic motion.
• Not being able to relate the solution of the differential equation to the amplitude, frequency, and period of simple harmonic motion.
• Model damped and forced oscillations using differential equations:
• Misunderstanding the concepts of damped and forced oscillations.
• Not being able to identify the correct differential equation that models damped or forced oscillations.
• Making mistakes when solving the differential equation for damped or forced oscillations.
• Not being able to relate the solution of the differential equation to the amplitude, frequency, and period of damped or forced oscillations.
• Model real-life situations using coupled first-order differential equations:
• Misunderstanding the concept of a coupled system of differential equations.
• Not being able to identify the correct coupled system of differential equations to model a given real-world situation.
• Making mistakes when solving coupled first-order differential equations.
• Not being able to interpret the solution of a coupled system of differential equations in the context of the real-world situation it models.

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