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 2x5 - 10x + 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.)

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dydt= f (t, y), y(t0 ) = y0

The existence theorem

The existence theorem (p. 66) basically says that if f(t, y) is continuous “near” (t0, y0), then the differential equation

has a solution “near” time t0.

Most of the functions we’ll see in this class are continuous (at least, most of the time!).

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dydt= f (t, y), y(t0 ) = y0

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 (t0, y0 ).

HOWEVER, there is no guarantee that the solution you have for (t0, y0 ) will be valid for times far from t0.

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.

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dydt= f (t, y) =1+ y2

Uniqueness of solutionsOK, 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 (t0, y0), a solution exists and is unique near (t0, y0). Otherwise, there might be more than one solution!

Try dy/dt = 3y2/3, y(0)=0, to see how this can look.€

∂f ∂y

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