The Domain of a Particular Solution

Separation of variables often hands you a formula like y = 1/(1-x) that is defined everywhere except one x-value. But the differential equation asked for a function that is smooth and continuous, with a derivative that exists at every point in its domain. At x = 1, the formula has a vertical asymptote: y shoots off to infinity, so no function value exists there and the curve cannot be continuous across it. That single bad point splits the formula's graph into two disconnected pieces, or branches, one for x < 1 and one for x > 1.

The initial condition is what tells you which branch is actually the solution. A solution curve is a connected path that starts at the initial point and follows the slope field outward. It cannot teleport across the asymptote to the other branch, because that would mean jumping over a point where the function is undefined. So you look at which side of the asymptote the initial point sits on, and keep only that side. The domain of the particular solution is an interval, bounded by the asymptote, not the full domain of the algebraic expression.

This idea closes out the separation of variables process: first solve for y, then check whether the resulting formula has any breaks, then use the initial condition to restrict the domain to the correct connected piece. It also deepens what a slope field already shows visually, since integral curves in a slope field often appear to run toward a vertical asymptote and stop, never crossing to the next branch.