stylistic:style_transfer, information:classification
Local existence and uniqueness theorem simplified

The theorem can be stated simply as follows. For the equation and initial value problem: y ′ = F (x, y), y 0 = y (x 0) {\displaystyle y'=F(x,y)\,,\quad y_{0}=y(x_{0})} if F and ∂ F /∂ y are continuous in a closed rectangle R = × {\displaystyle R=\times } in the x-y plane, where a and b are real (symbolically: a, b ∈ R) and × denotes the Cartesian product, square brackets denote closed intervals, then there is an interval I = ⊂ {\displaystyle I=\subset } for some h ∈ 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.