a linear initial value problem

A linear initial value problem is a first-order linear equation together with one starting condition: solve y' + p(x) y = q(x) given y(x_0) = y_0. As with any IVP you are after a single specific solution rather than the whole family — the unique curve that obeys the equation and passes through the given point.

The procedure is the integrating-factor recipe with a final substitution. Build mu(x) = e^(integral p dx), rewrite the equation as (mu y)' = mu q, integrate to get the general solution y = (1/mu)(integral of mu q dx + C), then plug in y(x_0) = y_0 and solve for C. Using definite integrals from x_0 makes this especially clean and pins C automatically. The headline fact is the contrast with the nonlinear case: if p and q are continuous on an open interval containing x_0, then the linear IVP has EXACTLY ONE solution, and that solution exists across the entire interval where p and q stay continuous. No surprises, no branching, no early collapse.

This guaranteed existence-and-uniqueness on a predictable interval is what makes linear models so trustworthy for prediction — circuits, cooling, mixing all inherit it. The boundary you should respect is that the solution is promised only on the interval where p and q are continuous: if p(x) = 1/x, an initial condition at x_0 = 1 gives a solution valid for x greater than 0, and that interval generally does NOT extend across the discontinuity at x = 0. The interval of validity is set by the coefficients, not by the solution running into trouble on its own.