Question

For the IVP, dy/dx= f(x,y), y(x0)= y0, the continuity of f(x,y) and ∂f/∂y guarantees the unique solution of the problem.
True or false with full explanation

Answer 1

Existence and Uniqueness of Solutions

Theorem: Let the function $f$ and $\partial f/\partial y$ be continuous in some rectangle $\alpha<x<\beta, \gamma<y<\delta,$ containing the point $(x_0, y_0).$ Then, in some interval $x_0-n<x_0<x_0+h$ contained in $\alpha<x<\beta$ there is a unique solution $y=\varphi(x)$ of the initial value problem

$$dy/dx=f(x, y), y(x_0)=y_0$$

The statement is true.