Differential equations are an important topic to be studied in A Level Maths. These equations model everything from the simplest rates of change to the complexities of physical phenomena. This guide is designed not just to explain what differential equations are but to simplify their study, making them accessible and achievable.
Differential equations are a type of equation that involves functions and their derivatives. They are powerful tools in theoretical and applied mathematics and can describe a huge number of scenarios. For example, they can show the growth of populations, the decay of radioactive materials and the oscillation of springs. Fundamentally, differential equations explore rates of change, showing how a system evolves over time or space.
We will show the essentials of differential equations - including the various types, from first order to non-linear - and provide you with the skills to tackle them. We have included quiz questions throughout to test your understanding and apply what you've learned in a practical context. This guide also serves as a valuable resource for teachers and parents looking to support their students or children in maths.
Types of Differential Equations in Maths
Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They tell us how a quantity changes over time or space. These equations are used in various fields (physics, engineering, economics, biology, etc) as they model the behaviour of complex systems ranging from the motion of planets to the growth of populations.
There are two main types of differential equations:
- Ordinary Differential Equations (ODEs): These use functions of a single variable and their derivatives. ODEs are used to model phenomena in a single-dimensional space, such as the cooling of a hot object in a room.
- Partial Differential Equations (PDEs): These involve functions of multiple variables and their partial derivatives. PDEs are used to model phenomena in multidimensional spaces, like the diffusion of pollutants in the atmosphere or the vibration of a drumhead.
First Order Differential Equations are a subtype of ODEs. They use only the first derivative of the unknown function. They are often the introductory topic for students learning about differential equations as they provide the foundation for understanding more complex equations. The general form of a first-order ODE is expressed as dy/dx=f(x,y) In this formula, y is the unknown function of x and f(x,y) is a function involving x and y.
Solving a differential equation means finding the function y that satisfies the equation for given conditions. This process involves integrating the function, applying methods to the equation's form and considering initial or boundary conditions.
1
Which of the following is an example of a First Order Differential Equation?
Solving different types of Differential Equations
It's important to understand the different types of differential equations and how to solve them. This will help you to tackle a wide range of problems in physics, engineering and beyond.
- First Order Differential Equations are the simplest form, where the equation uses only the first derivative of the function. These equations often model rates of change in one variable and can be solved using methods like separation of variables or integrating factors. For example, equations describing exponential growth or decay, such as population dynamics.
- Second Order Differential Equations involve the second derivative of the function and are pivotal in describing systems with acceleration, such as mechanical vibrations or electrical circuits. The solution to these equations can vary depending on whether the equation is homogeneous or non-homogeneous and if specific coefficients are constant or variable.
- Homogeneous Equations have solutions that can be added together to form another solution. They often appear in scenarios where the rate of change is proportional to the function's current value, leading to exponential growth or decay solutions.
- Non-Homogeneous Equations include an additional function that does not equal zero. These equations are common in applied mathematics and model processes where external forces or sources influence the system.
- Linear Differential Equations are a subset where the unknown function and its derivatives appear linearly. These are particularly important because they have well-established solution techniques, including the superposition principle for homogeneous linear equations.
- Non-Linear Differential Equations do not show linearity, making them more complex to solve. They describe a wide array of phenomena, from chaotic systems in weather patterns to nonlinear dynamics in population genetics.
2
What makes a homogeneous differential equation different to a non-homogeneous one?
- Consider the linear differential equation 2y′′+3y′−y=0. This is homogeneous because it can be set to zero (the right side of the equation is zero). If we compare it to a non-homogeneous equation like 2y′′+3y′−y=sin(x), the presence of sin(x) on the right side introduces an external influence, making it non-homogeneous.
Basic concepts and terminology in Differential Equations
There are foundation concepts and terminology that we need to understand before learning about different equations in more detail.
- Function and Derivative: The function is crucial to differential equations, which describe a relationship between two variables, typically x (the independent variable) and y (the dependent variable). The derivative, denoted as dx/dy​, represents the rate at which y changes with respect to x.
- Order of a Differential Equation: The order is decided by the highest derivative present in the equation. For example, an equation involving d2y / dx2​ is a second-order differential equation because the second derivative is the highest derivative in the equation.
- Degree of a Differential Equation: The degree is defined by the power of the highest derivative, assuming the equation is polynomial in derivatives. For example, if the highest derivative appears squared, the equation's degree is two.
- Linear vs. Non-Linear Differential Equations: A differential equation is linear if it can be written so that each term is a constant or a product of a constant and a single variable or its derivative. Non-linear differential equations involve terms that are products of variables and derivatives, powers of derivatives, or functions of the dependent variable or its derivatives.
- Initial Conditions: These are values specified for the solution at a particular point, used to find a specific solution to a differential equation from the family of possible solutions. For first-order differential equations, one initial condition is usually given (y at a specific x), while second-order equations typically require two initial conditions.
3
What does the order of a differential equation tell us?
The solution to First Order Differential Equations
First Order Differential Equations come in various forms and each requires a specific approach to be solved.
- Separation of Variables: This method is used when a differential equation can be rearranged so all terms involving the dependent variable (usually y) are on one side of the equation and all terms involving the independent variable (usually x) are on the other side. The equation can be integrated with respect to each variable separately. This technique is particularly useful for equations that model natural growth and decay processes, as well as simple mixing problems.
- Integrating Factor Method: Used mainly for linear first-order differential equations of the form dx / dy​+P(x)y=Q(x), where P(x) and Q(x) are functions of x alone. The integrating factor, usually denoted as I(x), is an exponential function derived from P(x) that, when multiplied by the original equation, allows the left-hand side to be expressed as the derivative of a product, which allows integration on both sides.
- Exact Equations: An exact differential equation can be written in the form M(x,y)dx+N(x,y)dy=0, where M and N have continuous partial derivatives and the equation satisfies the condition aM / ay = aN / ax​. Solving exact equations requires finding a function ψ(x,y) whose total differential equals the left-hand side of the equation, effectively reducing the problem to a simpler integration task.
Each of these methods offers a pathway to solve first-order differential equations, emphasising the importance of understanding the equation's structure and characteristics before selecting the appropriate solving technique.
4
Which method is best for a first-order linear differential equation of the form dx / dy​+P(x)y=Q(x)?
Second Order Differential Equations
Learning about Second Order Differential Equations is a significant step forward in A Level Maths. These equations, which involve the second derivative of the unknown function, are pivotal in modelling systems where acceleration or curvature is a factor, such as in physics and engineering.
Homogeneous Linear Second Order Differential Equations: These equations have the general form a d2y / dx2 + b dy / dx2 + cy = 0, where a, b and c are constants. A key strategy in solving these equations involves finding the roots of the characteristic equation ar2+br+c=0. The nature of the roots (real and distinct, real and repeated, or complex) dictates the form of the solution, which can be a combination of exponential functions, sine and cosine functions, or a mix of both.
Non-Homogeneous Linear Second Order Differential Equations: These equations include an additional term that makes them non-homogeneous, such as a d2y / dx2 + b dy / dx + cy = g(x), where g(x) is a known function. Solving these involves finding a particular solution that satisfies the non-homogeneous term and then adding it to the general solution of the corresponding homogeneous equation. Methods to find the particular solution include the method of undetermined coefficients and the method of variation of parameters.
Reduction of Order: This technique is useful when one solution of a second-order differential equation is known and it allows for the determination of a second, linearly independent solution. The process involves reducing the second-order equation to a first-order equation by substituting a variable for a derivative of the solution.
To understand and apply these methods, we need a solid grasp of algebra and calculus fundamentals, including characteristic equations, integration and the concept of linear independence. Mastery of these techniques allows us to solve complex problems in theoretical and applied mathematics.
5
What method is normally used to solve a homogeneous linear second order differential equation with constant coefficients?
Exploring Homogeneous and Non-Homogeneous Equations
Understanding the difference between homogeneous and non-homogeneous equations is important as it allows us to find the best solution strategy.
Homogeneous Differential Equations: These equations are characterised by their lack of a term independent of the unknown function and its derivatives. In simpler terms, a differential equation is homogeneous if every term is a function of the dependent variable y and its derivatives. Homogeneous equations can often be solved by finding a set of solutions that can be multiplied by a constant to return another solution to the same equation. This property leads to solutions that often involve arbitrary constants, reflecting the equation's inherent symmetry.
For first-order homogeneous differential equations, the standard form is dx / dy​=f(xy​), where the function f is a function of the ratio xy​. Solving these equations typically involves substitution techniques that transform the differential equation into a separable equation, allowing for integration and solution.
Non-Homogeneous Differential Equations: These equations include at least one term that isn't a function of the unknown function y or its derivatives. This additional term is known as the 'forcing function' and introduces an external influence into the system being modelled. The presence of the forcing function requires finding a particular solution that directly addresses this term, in addition to the general solution of the corresponding homogeneous equation.
The method of undetermined coefficients and the method of variation of parameters are commonly used to find particular solutions to non-homogeneous equations. The choice between these methods depends on the form of the forcing function. The method of undetermined coefficients is most suitable for forcing functions that are polynomials, exponentials, sines and cosines. The method of variation of parameters can be applied more broadly.
6
What key feature differentiates a non-homogeneous differential equation from a homogeneous one?
Linear and Non-Linear Differential Equations
Linear Differential Equations: These equations are characterised by the dependent variable and all its derivatives appearing linearly, meaning each term is either a constant or a product of a constant and the dependent variable or its derivatives. The general form of a first-order linear differential equation is dx / dy​+ P(x)y=Q(x), where P(x) and Q(x) are functions of x only. For higher-order linear differential equations, the form extends to an​(x)dny/dxn​+an−1​(x)dn−1y/dxn-1​+⋯+a1​(x)dxdy​+a0​(x)y=G(x), where G(x) is the non-homogeneous term.
Linear differential equations are important because they have clear solution strategies and superposition, which applies to homogeneous linear differential equations. This superposition principle says that if y1​(x) and y2​(x) are solutions, any linear combination of these solutions, c1​y1​(x)+c2​y2​(x), is also a solution.
Non-Linear Differential Equations: These equations use the dependent variable or its derivatives in a non-linear manner, such as being multiplied together, raised to a power, or appearing inside a function (e.g. sin(y)). These equations don't satisfy the superposition principle and are more complex. Non-linear differential equations are often seen in advanced physics and engineering contexts, modelling phenomena such as turbulent fluid flow, nonlinear oscillations and population dynamics in biology. Solving these equations requires sophisticated analytical techniques, numerical methods and sometimes simulations to understand the behaviour of solutions over time.
The challenge with non-linear differential equations lies not just in finding solutions but in the analysis of their stability, periodicity and sensitivity to initial conditions, which can lead to phenomena like chaos in certain systems.
7
Why are non-linear differential equations considered more complex to solve than linear differential equations?
Practical applications and examples of Differential Equations
Differential equations are crucial in modelling and solving real-world problems across various disciplines. By applying these concepts and methods, we can explore how differential equations enable us to understand and predict the behaviour of complex systems in science, engineering and beyond.
- Physics and Engineering: Differential equations are foundational in physics for modelling motion, energy and waves. For example, Newton's second law of motion can be expressed as a differential equation. In engineering, the design of electrical circuits often relies on differential equations to describe the relationship between voltage, current and resistance over time. To see differential equations in action, explore PhET Interactive Simulations, which offer a range of science and mathematics simulations to enhance your understanding of related principles.
- Biology and Medicine: Differential equations are used in biology to model population dynamics through the Lotka-Volterra equations for predator-prey interactions or the logistic growth equation for population limits. In medicine, they model the spread of diseases, the dynamics of immune response, or the rate of drug absorption and elimination in the body.
- Economics and Finance: The Black-Scholes equation, a partial differential equation, is a cornerstone in financial mathematics for option pricing. In economics, differential equations model the growth of economies, the impact of interest rates on investment, or the dynamics of markets and competition.
- Environmental Science: Climate models extensively use differential equations to simulate atmospheric and oceanic processes, predicting climate change impacts. Differential equations also model pollutant dispersion in air and water, helping in environmental protection efforts.
For visualizing solutions and exploring the behaviour of functions and their derivatives, Wolfram Alpha offers a comprehensive equation solver.
8
Which field uses differential equations to model predator-prey interactions and population dynamics?
Wrapping up
Differential equations are powerful tools that allow us to model, understand and predict the world around us. From the oscillation of a pendulum to the growth of populations and from the flow of currents in electrical circuits to the valuation of financial derivatives, differential equations serve as the language of change and evolution across numerous disciplines.
For A Level Maths students, mastering differential equations lets us explore advanced studies in mathematics, physics, engineering and beyond. It equips you with a set of analytical skills that are highly valued in problem-solving and critical thinking.
Here are some suggestions for getting to grips with differential equations:
- Engage with real-world problems: Apply what you've learned by modelling real-world phenomena. Start with simple models and gradually increase complexity.
- Utilise technology: Software tools and online platforms offer valuable resources for visualising differential equations and their solutions. Experiment with graphing tools and numerical solvers to gain deeper insights.
- Participate in math communities: Join online forums, math clubs, or study groups where you can discuss concepts, share insights and solve problems collaboratively.
- Seek out advanced topics: Once you're comfortable with the basics, explore more advanced topics in differential equations, such as nonlinear dynamics, chaos theory, or partial differential equations.
As you continue your studies and professional development, remember that the complexity of differential equations mirrors the complexity of the world they help to describe. If you need more support understanding differential equations, you may consider searching our list of verified A Level Maths tutors.
