Exercise 1

Which of the following initial value problems are guaranteed to have a unique solution by the Existence and Uniqueness Theorem (Existence and Uniqueness Theorem)? In each case, justify your conclusion.

1. $y' = y^2 + y^3$, $y(0) = 1$
2. $y' = \sqrt[4]{y}$, $y(1) = 0$
3. $y' = \sqrt[4]{y}$, $y(1) = 1$
4. $x' = \dfrac{t}{x^2 - 4}$, $x(0) = 2$
5. $x' = \dfrac{t}{x^2 - 4}$, $x(2) = 0$
6. $y' = x \sin y$, $y(0) = 0$
7. $y' = \dfrac{1}{t - 1} y + 2t$, $y(1) = 1$