A differential equation is an equation that relates a function with one or more of its derivatives. It describes how a quantity changes over time or space. For example, dy dx equals 2x tells us that the rate of change of y equals 2x. The solution to this equation is y equals x squared plus C, where C is a constant.
Differential equations can be classified in several ways. By order, we have first order equations involving dy dx, and second order equations involving d squared y dx squared. By linearity, we have linear equations where coefficients are constants or functions of x only, and nonlinear equations involving products or powers of y and its derivatives. Here are examples of each type.
There are several methods for solving differential equations. Separation of variables is one of the most common techniques. For example, given dy dx equals xy, we separate variables to get dy over y equals x dx. Then we integrate both sides to get ln absolute value of y equals x squared over 2 plus C. This gives us a family of solution curves as shown in the graph.
Differential equations have countless applications across many fields. In physics, they describe motion through Newton's second law and oscillations. In biology, they model population growth and radioactive decay. In engineering, they analyze electrical circuits and heat transfer. The graphs show examples of exponential growth and decay, which are solutions to simple first-order differential equations.
To summarize what we have learned about differential equations: They are mathematical equations that relate functions to their derivatives, describing how quantities change. We can classify them by order and linearity, solve them using various methods like separation of variables, and apply them across many fields to model real-world phenomena involving change and motion.