Chapter 1.5: Existence and Uniqueness of Solutions

Uniqueness and qualitative analysis

In some cases, we can use the Uniqueness Theorem and some qualitative information to give more exact information about solutions. Not only can we determine the range of the solution, we can determine the domains and how it affects the solution among equilibrium points.

Uniqueness Theorem

Suppose f(t,y) and δf/δy are continuous functions in a rectangle of the form {(t,y) | a<t<b, c<y<d} in the ty-plane. If (t₀,y₀) is a point in this rectangle and if y₁(t) abd t₂(t) are two functions that solve the initial-value problem: dy/dt=f(t,y), y(t₀)=y₀ for all t in the interval t₀-ε<t<t₀+ε where ε is some positive number, then: y₁(t)=y₂(t) for t₀-ε<t<t₀+ε. That is, the solution to the initial-value problem is unique.

Existence theorem

Suppose f(t,y) is a continuous function in a rectangle of the form {(t,y) | a<t<b, c<y<d} in the ty-plane. If (t₀,y₀) is a point in this rectangle, then there exists an ε>0 and a function y(t) defined for t₀-ε<t<t₀+ε that solves the initial-value problem: dy/dt=f(t,y), y(t₀)=y₀ This theorem says that as long as the function on the right-hand side of the differential equation is reasonble, solutions exist. It doesn't rule out the possibility that solutions exist even if f(t,y) is not a nice function, but it doesn't guarantee it either.