Welcome to an introduction to differential equations. A differential equation is an equation that relates a function with one or more of its derivatives. For example, the equation dy dx equals 2x is a differential equation. Here, we can see the function y equals x squared in blue, and its derivative, y prime equals 2x, in red. At any point on the curve, the derivative gives the slope of the tangent line, as shown by the green line at x equals 1.
Differential equations can be classified in several ways. Ordinary Differential Equations, or ODEs, involve derivatives with respect to a single variable. For example, this is a second-order ODE with the highest derivative being d squared y over d x squared. Partial Differential Equations, or PDEs, involve partial derivatives with respect to multiple variables, like this heat equation. The order of a differential equation is determined by the highest derivative that appears in the equation. Here we have examples of first-order and second-order ODEs.
Let's explore some common methods for solving differential equations. One fundamental technique is separation of variables, where we rearrange the equation to separate the variables on different sides. For example, with the equation dy dx equals x y, we can separate the variables by dividing both sides by y. This gives us 1 over y dy equals x dx. We then integrate both sides, resulting in the natural log of the absolute value of y equals x squared over 2 plus a constant. Solving for y, we get y equals C times e to the power of x squared over 2. The graph shows solutions for different values of the constant C. Other methods include using an integrating factor and numerical approaches like Euler's method.
Differential equations have numerous real-world applications across various fields. In physics, they model motion, vibration, heat transfer, and electromagnetic phenomena. Engineers use them for circuit analysis, structural design, and control systems. In biology, differential equations help understand population dynamics, disease spread, and neural activity. This graph shows two population growth models: exponential growth in blue, where the population grows without bounds, and logistic growth in red, which accounts for limited resources with a carrying capacity. The logistic growth is modeled by this differential equation. In economics, differential equations model market equilibrium, economic growth, and option pricing. These applications demonstrate why differential equations are fundamental tools in science and engineering.
To summarize what we've learned about differential equations: A differential equation relates a function with one or more of its derivatives. They can be classified as Ordinary Differential Equations, which involve a single independent variable, or Partial Differential Equations, which involve multiple independent variables. The order of a differential equation is determined by the highest derivative that appears in the equation. Common solution methods include separation of variables, integrating factors, and numerical approaches like Euler's method. Differential equations have widespread applications across various fields including physics, engineering, biology, and economics, making them essential tools for modeling real-world phenomena where rates of change are involved.