Section 1.5 EXISTENCE AND UNIQUENESS OF SOLUTIONS

The eternal existential question If we are given any old initial-value problem Does there have to be a solution? If so, could there be more than one solution? (Think of questions like “does 2x x + 5 = 0 have a solution? if so, how many?” We can show that there is a solution between x=-1 and x=1, but we can’t factor the polynomial to find it, and we don’t know how many there are.)

The existence theorem The existence theorem (p. 66) basically says that if f(t, y) is continuous “near” (t 0, y 0 ), then the differential equation has a solution “near” time t 0. Most of the functions we’ll see in this class are continuous (at least, most of the time!).

Formal statement of the existence theorem Check out the theorem on p. 66. The statement “there exists an  > 0” means that there is some positive value the variable  can take on so that the statement becomes true. The theorem does not tell us how large that value is.

Example This is a continuous function, so solutions will exist “near” any point (t 0, y 0 ). HOWEVER, there is no guarantee that the solution you have for (t 0, y 0 ) will be valid for times far from t 0. For example, the general solution is y(t)=tan(t+c), so a solution of the IVP y(0)=0 is y(t) = tan(t), which is undefined at t = pi/2. This situation is called lack of extendability. The existence theorem doesn’t guarantee extandability.

Uniqueness of solutions OK, so in most reasonable situations, at least one solution to an IVP will exist. Did I say at least one???? Does this mean there can be more than one???? YES. If the function f(t, y) and its partial derivative are continuous at (t 0, y 0 ), a solution exists and is unique near (t 0, y 0 ). Otherwise, there might be more than one solution! Try dy/dt = 3y 2/3, y(0)=0, to see how this can look.

Questions What is the difference between the Existence Theorem and the Uniqueness Theorem? In particular, how do their hypotheses differ? What does this tell us about how solution curves “look?” (There’s a particularly nice relationship for equilibrium solutions.) How can these theorems help us to spot problems in numerical solutions? Exercises: p. 73: 2, 13