Finding Particular Solutions Using Initial Conditions

Separating variables and integrating both sides of a differential equation gives you a general solution loaded with an unknown constant, usually called C. That constant isn't junk left over from integration — it's the whole reason the solution describes infinitely many curves instead of one. Each value of C shifts or stretches the curve, so the family covers every possible curve with that slope behavior. An initial condition is one ordered pair (x0, y0) the actual curve must pass through. Plug those numbers into the general solution in place of x and y, and the equation now has only one unknown: C. Solve for it, substitute back, and you've traded the whole family for the single particular solution that fits your starting point.

The mechanism is just substitution and algebra, but order matters: integrate first, find the +C, and only then use the initial condition. Don't try to use the condition before separating and integrating — there's no constant yet to pin down. Once C is a number, the particular solution is a normal function, and you can evaluate it anywhere, graph it, or use it to predict a value, exactly like evaluating any other function.

This caps off the separation-of-variables process from earlier in the unit: separate, integrate, add C, then use the given point to finish the job. It also connects forward to modeling problems — exponential growth and decay, Newton's law of cooling, logistic growth — where the differential equation describes a rate and the initial condition is the real-world starting value, like an initial population or temperature.