Existence and Uniqueness Theorem for ODE’s The following is a key theorem of the theory of ODEs; it has immediate (and crucial) consequences for sketching integral curves. Theorem. Existence and Uniqueness Theorem for ODEs For any ( a, b) in the region where f is deﬁned, y� = f ( x, y) has exactly one solution such that y( a) = b. Remark 1. The solution y( x ) needn’t be deﬁned for all x: it may only exist for x near a. Remark 2. You actually have to put some technical conditions on f . The most important one is that the theorem can fail if f ( x, y) is not continuous at ( a, b). That is, there might be many solution curves or no solution curves that pass through the point. By the existence part of the theorem, there is an integral curve through any point where f ( x, y) is deﬁned. Now suppose you have two integral curves through the same point. By the uniqueness part of the theorem, they must agree: Theorem. Intersection Principle Integral curves cannot intersect; every point lies on exactly one integral curve. This principle is key in sketching integral curves on a direction ﬁeld: you should never draw two that intersect, i.e. they can’t cross or even touch at a point. We will see plenty of illustrations of the principle using the Isoclines applet.