First-order differential equations

An algebraic equation asks which number satisfies a relation. A differential equation asks which function satisfies a relation involving its own rate of change. That shift is the whole difficulty and the whole power: nature is full of statements of the form "the rate at which this changes depends on how much of it there is", and those are differential equations rather than algebraic ones.

The simplest useful type is separable: get all the y's on one side with dy and all the x's on the other with dx, then integrate both sides. dy/dx = 2y becomes dy/y = 2 dx, integrates to ln|y| = 2x + C, and rearranges to y = Ce^(2x). Notice what came out — not a number, but a family of functions, one for each C. Exponential growth and decay, Newton's law of cooling and radioactive decay all arrive this way. The linear first-order equation dy/dx + P(x)y = Q(x) is the other standard type, solved by multiplying through by an integrating factor chosen to make the left side a single derivative.

The constant is where the physics enters. A rate of change cannot know where the process started, so the equation alone gives the whole family; an initial condition such as y(0) = 5 picks out one member. Without it the answer is genuinely incomplete rather than merely untidy. And a warning worth carrying forward: separable and linear equations are the well-behaved minority. Most differential equations have no solution expressible in elementary functions at all, which is why numerical methods and qualitative analysis matter as much as solving techniques.