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The Existence and Uniqueness Theorem

DIFFEQ-MWLE4A

Consider the general first order differential equation:

$$$$\frac{dy}{dt} = f(t,y), y(0) = 0$$$$

Which of these statements concerning existence and uniqueness of $y(t)$ are true?

A

If $f$ and ${\partial f \over \partial y}$ are continuous in a rectangle $|t|\leq a$ and $|y| \leq b$, then there is some interval $|t| \leq h \leq a$ in which there exists a unique solution

B

If ${\partial f \over \partial y}$ is continuous in a rectangle $|t|\leq a$ and $|y| \leq b$, then there is some interval $|t| \leq h \leq a$ in which there exists a unique solution

C

If $f$ is continuous in a rectangle $|t|\leq a$ and $|y| \leq b$, then there is some interval $|t| \leq h \leq a$ in which there exists a unique solution

D

If $f$ and ${\partial f \over \partial y}$ are continuous in a rectangle $|t|\leq a$ and $|y| \leq b$, then there exists a unique solution for all $-\infty < t < \infty$