SQUARE45

The theorem can be stated simply as follows. For the equation and initial value problem:

y'=F(x,y)\,,\quad y_{0}=y(x_{0})

F

and

\partial F/\partial y

are continuous in a closed rectangle

R=[x_{0}-a,x_{0}+a]\times [y_{0}-b,y_{0}+b]

in the

x-y

plane, where

a

and

b

are real (symbolically:

a,b\in \mathbb {R}

) and

\times

denotes the Cartesian product, square brackets denote closed intervals, then there is an interval

I=[x_{0}-h,x_{0}+h]\subset [x_{0}-a,x_{0}+a]

for some

h\in \mathbb {R}

where the solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique. Since there is no restriction on

F

to be linear, this applies to non-linear equations that take the form

F(x,y)

, and it can also be applied to systems of equations. - ^Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ISBN0-471-83824-1

Local existence and uniqueness theorem simplified

The statement of the theorem