Discussion
The initial value problem :
\[ \frac{dy}{dx} = \sqrt{|y|},\quad y(0) = 0, \]
Answer: D
The problem has more than one solution.
\(y \equiv 0\) is a solution. So is
\[ y(x) = \begin{cases} x^{2}/4, & x \ge 0 \\ 0, & x \lt 0 \end{cases} \]
since for \(x \ge 0\), \(y' = x/2 = \sqrt{x^{2}/4} = \sqrt{|y|}\), and \(y(0) = 0\). In fact infinitely many solutions branch off the zero solution at any point \(c \ge 0\).
Uniqueness fails because \(\sqrt{|y|}\) is not Lipschitz in y near \(y = 0\). So 'no solution' and 'unique solution' are wrong, and 'has non-zero solution' understates what happens.
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