Calculus is the mathematics of change and motion. It consists of two main branches: differential calculus and integral calculus. Differential calculus focuses on rates of change, represented by derivatives, which give us the slope of a curve at any point. Integral calculus deals with accumulation and finding areas under curves, helping us calculate total change over an interval. These two branches are connected by the Fundamental Theorem of Calculus.
Differential equations are mathematical equations that relate a function with its derivatives. They are used to model systems that change over time or space. For example, the equation dy/dx equals k times y describes exponential growth or decay, where the rate of change of a quantity is proportional to the quantity itself. This simple equation models many natural phenomena like population growth, radioactive decay, and compound interest. The solution to this differential equation is y equals y-sub-zero times e raised to the power of k times t, where k determines whether the function grows or decays.
To solve differential equations, we use various methods depending on the equation type. One common approach 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 times y, we can separate it to get 1/y dy equals x dx. Integrating both sides gives us ln of absolute y equals x squared over 2 plus C. Solving for y, we get y equals C times e raised to x squared over 2. This solution represents a family of curves, where different values of C give different specific solutions. The vector field shows the direction of the rate of change at each point, and the solution curves follow these directions perfectly.
Differential equations have countless applications across various fields. In physics, they describe motion, waves, and heat transfer. In biology, they model population growth, like the logistic growth model shown here, which accounts for limited resources through a carrying capacity. The equation dP/dt equals k times P times (1 minus P over K) describes how population growth slows as it approaches the carrying capacity. In engineering, differential equations analyze circuits and control systems. In economics, they model market dynamics and financial instruments. These applications demonstrate how differential equations provide a powerful framework for understanding and predicting change in complex systems.
To summarize what we've learned: Calculus provides the fundamental tools for studying change, with differential calculus focusing on rates of change through derivatives, and integral calculus examining accumulation through integrals. Differential equations build on these concepts by relating functions to their derivatives, allowing us to model how systems evolve over time or space. We can solve these equations using various methods, including separation of variables, integrating factors, and numerical approaches. The applications of differential equations are vast, spanning physics, biology, engineering, economics, and countless other fields. This powerful connection between calculus and differential equations gives us the mathematical framework to understand, model, and predict complex systems in our world.