Concept preview

What you will be able to do

1. 01Integrate a separable differential equation to obtain a GENERAL solution and understand why the constant of integration $+C$ makes it a whole family of curves.
2. 02Apply an initial condition to solve for $C$ and write the single PARTICULAR solution that passes through the given point, then evaluate it at a later input.
3. 03Determine the interval on which a particular solution is valid (its domain of validity), and avoid the classic error of applying the initial condition before integrating.

A differential equation such as $\dfrac{dy}{dx}=t$ (or $\dfrac{dy}{dx}=x$) does not have a single solution; it has a whole family of them. Antidifferentiating gives $y=\dfrac{x^2}{2}+C$, and every choice of the constant $C$ slides the parabola up or down to produce a different valid solution. We call $y=\dfrac{x^2}{2}+C$ the GENERAL solution because it captures all of them at once. The constant $C$ is not a nuisance to be dropped — it is the dial that selects which member of the family we mean.