1.Consider the differential equation $\displaystyle \frac{dy}{dx}=\sqrt{x-y};\ y(2)=2$.

$\displaystyle f(x, y)=\sqrt{x-y}$ and $\displaystyle D_yf(x, y)=-\frac{1}{2\sqrt{x-y}}$

Neither $\displaystyle f(x, y)$ nor $\displaystyle D_yf(x, y)$ is continuous on some rectangle that contains (2, 2) in its interior. Neither existence nor uniqueness is guaranteed in any neighbourhood of x = 2.

2.Consider the differential equation $\displaystyle \frac{dy}{dx}=2\sqrt{y};\ y(0)=0$.

$\displaystyle f(x, y)=2\sqrt{y}$ and $\displaystyle D_yf(x, y)=\frac{1}{\sqrt{y}}$

Here also, neither $\displaystyle f(x, y)$ nor $\displaystyle D_yf(x, y)$ is continuous on some rectangle that contains (0, 0) in its interior. However, two different solutions exist: $\displaystyle y_1(x)=x^2$ and $\displaystyle y_2(x)\equiv 0$. Can someone please explain this anomaly?