Existence & Uniqueness

Theorem: If is a differential equation such that is continuous in the region , and if there exists a constant such that

(1.6)

For all and all (this is called the Lipschitz condition and is called the Lipschitz constant), then there exists a unique continuously differentiable function such that

and , the initial condition.

An example that illustrates the fact that an initial value problem may have no solution is the equation

The family of solutions is . Since all solution pass through , initial values cannot be given at the singular point .