Consider a first order differential equation
with the initial condition
where is bounded in the neighborhood of the initial point, i.e.,
in
Sufficient Condition of Existence: If is continuous in the neighborhood region , the solution of this initial value problem in the region exists.
Sufficient Condition of Existence and Uniqueness: If and its partial derivative with respect to are continuous in the neighborhood region , the solution of this initial value problem in the region exists and is unique.
Picard Iteration Method: The unique solution of the above initial value problem is
where
