interval of validity of a solution

A solution of a differential equation is not automatically a function defined for all x — it lives on some interval, and stepping outside that interval the formula may stop making sense or stop solving the equation. The interval of validity is the largest open interval, containing the initial point, on which the solution exists and is genuinely a differentiable solution. It answers: for what range of x is this answer actually good?

Two different things can limit the interval. First, the equation's coefficients: for a linear IVP y' + p(x) y = q(x), the solution is guaranteed only across the open interval where p and q are both continuous, and it cannot in general be pushed past a discontinuity. If p(x) = 1/x and the initial point is x_0 = 2, the interval of validity is x greater than 0 — the singularity at x = 0 forms a wall. Second, the solution itself can fail even when the equation is smooth: a nonlinear solution like y = 1/(1 - x) is fine until x = 1, where it blows up to infinity, so its interval of validity stops at 1 regardless of how nice f looked. You read off the interval by finding the nearest place, on each side of x_0, where the solution becomes undefined, discontinuous, or unbounded.

This idea keeps you honest about how far a model's prediction can be trusted. For linear problems the interval is determined in advance by the coefficients — a real convenience. For nonlinear problems it can depend on the initial condition and may be much shorter than expected, which is why 'find the interval of validity' is a standard part of solving an IVP, not an afterthought. The common mistake is to report an algebraic formula and assume it holds everywhere; it holds only on the connected piece around the initial point.