Solving a differential equation isn’t just about crunching numbers—it’s about finding a function that makes the equation true. In this chapter, we’ll explore what it really means for something to be a “solution” to a differential equation and why that idea is more subtle (and more powerful) than it might first appear.
We’ll look at different kinds of solutions—general solutions, which describe an entire family of possibilities, and particular solutions, which pin down a single curve using initial conditions. Along the way, we’ll see how integration, one of the main tools from calculus, becomes our first method for actually solving a simple differential equation.
By the time you finish this chapter, you’ll be able to test whether a function is truly a solution, describe the different types of solutions we use, and solve the simplest cases by integrating. These skills will become the groundwork for every solving technique that follows.
