Welcome to differential equations. A differential equation relates a function with its derivatives, describing how quantities change over time or space. These equations are fundamental in modeling real-world phenomena. For example, Newton's Second Law of Motion can be expressed as a differential equation where force equals mass times acceleration, which is the second derivative of position with respect to time. Here we see a simple second-order differential equation and one of its solutions, showing damped oscillation.
Differential equations are classified into different types. The two main categories are Ordinary Differential Equations, or ODEs, and Partial Differential Equations, or PDEs. ODEs involve derivatives with respect to a single variable, like time or position. For example, a first-order linear ODE relates a function and its first derivative, while a second-order ODE like the simple harmonic motion equation involves second derivatives. PDEs, on the other hand, involve partial derivatives with respect to multiple variables. The Laplace equation and heat equation are common examples of PDEs used in physics and engineering. These different types of differential equations are used to model various physical phenomena across sciences.
Let's explore methods for solving first-order ordinary differential equations. Two common approaches are separation of variables and the integrating factor method. Separation of variables works when we can rearrange the equation to have all terms with y on one side and all terms with x on the other. For example, with the equation dy/dx equals xy, we can separate it as dy over y equals x dx. Integrating both sides gives us the natural logarithm of the absolute value of y equals x squared over 2 plus a constant. Solving for y, we get y equals C times e raised to x squared over 2. This gives us a family of solutions depending on the value of the constant C, as shown in the graph. Each curve represents a solution for a different value of C.
Differential equations have countless applications across various fields. In physics, they describe motion, waves, heat flow, and electromagnetic fields. In biology, they model population growth, predator-prey relationships, and the spread of diseases. Engineers use them for analyzing circuits, vibrations, fluid dynamics, and control systems. Even economists rely on differential equations to understand market dynamics and investment growth. Let's look at two examples: First, population growth can be modeled using either exponential growth, where the rate of change is proportional to the current population, or logistic growth, which accounts for limited resources. Second, a harmonic oscillator, like a spring-mass system, can be described by a second-order differential equation. The undamped oscillator maintains constant amplitude, while the damped oscillator gradually decreases in amplitude over time due to friction or resistance.
To summarize what we've learned about differential equations: First, they are mathematical equations that relate functions with their derivatives, describing how quantities change over time or space. Second, they are classified into ordinary differential equations, which involve a single independent variable, and partial differential equations, which involve multiple independent variables. Third, we can solve differential equations using various methods, including separation of variables and the integrating factor method for first-order linear equations. Fourth, differential equations have countless applications across diverse fields including physics, biology, engineering, and economics. Finally, they serve as fundamental tools for mathematical modeling, allowing us to understand and predict the behavior of complex systems in the real world.