Find out whether the following IVP satisfy the Picard-Lindelöf theorem

Question

We're asked to find out whether the following IVP satisfy the Picard-Lindelöf theorem

1. $\dot{y}(t)= (y(t))^{1/3}, y(0)=1$
2. $\dot{y}(t)= \sqrt[3]{(y(t)-1)^2}, y(0)=-1$
3. $\dot{y}(t)= 3 \sqrt[3]{(y(t)-1)^2}, y(0)=1$
4. $\dot{y}(t)= e^{\frac{1}{y(t)-2}}, y(4)=2$
5. $\dot{y}(t)= e^{\frac{1}{y(t)-2}}, y(4)=-1$

Answer 1

To satisfy the theorem, we need to check whether the function is continuous, and whether it is continuously differentiable, around the initial value.

1. The function is continuous around 1, and its derivative, $\frac{1}{3y^\frac{2}{3}}$, is also continuous around $1$. Thus the theorem holds.

2. The function is continuous around $y(0)=-1$, and its derivative $\dfrac{2}{3\sqrt[3]{y-1}}$ is also continuous around $y=-1$. Thus the theorem holds.

3. The function and its derivative are not continuous around $y=1$. Thus the theorem does not hold.
4. The function and its derivative are not continuous around $y=2$. Thus the theorem does not hold.
5. The function and its derivative -$\dfrac{\mathrm{e}^\frac{1}{y(t)-2}}{\left(y(t)-2\right)^2}$ are continuous around $y=-1$. Thus the theorem holds.