Existence and uniqueness theorem

The idea is simple: if \(f(x,y)\) is smooth on a region, the initial point \((x_0,y_0)\) fixes a direction at each instant and the solution curve is determined step by step. The precise conditions are that \(f\) and \(\partial f/\partial y\) are continuous on a rectangle containing the point. The result is local: the solution exists on an interval around \(x_0\), which may be small.

To see that «local» is real, consider \(y'=y^2\) with \(y(0)=1\). Both \(y^2\) and \(2y\) are continuous on the whole plane, and the solution is \(y=1/(1-x)\), which blows up at \(x=1\). The equation is perfectly smooth, and yet the solution does not live on the whole line.

When the hypothesis on \(\partial f/\partial y\) fails, uniqueness can be lost. The classic case is \(y'=\sqrt{y}\) with \(y(0)=0\): both \(y=0\) and \(y=x^2/4\) (for \(x\ge 0\)) solve the problem, because \(\partial f/\partial y=1/(2\sqrt{y})\) is not continuous at \(y=0\). A common mistake is to read this backwards: if the hypotheses fail, the theorem does not say there are several solutions, only that it concludes nothing.

A useful geometric consequence is that, under the hypotheses, two distinct solution curves never cross or touch. This allows, for example, arguing that a solution starting between two constant solutions stays trapped between them. The next concept, the direction field, draws exactly those curves without solving the equation.