The problem asks to investigate the existence and uniqueness of the solution to the initial value problem:

$\frac{dy}{dx} = |y|^{3/4}, \quad y(0) = 0.$

This is a first-order differential equation. To analyze the existence and uniqueness, we can apply Picard-Lindelöf Theorem (also known as the existence and uniqueness theorem), which gives conditions under which a unique solution exists near an initial point.

Picard-Lindelöf Theorem:

For the initial value problem $\frac{dy}{dx} = f(x, y)$, $y(x_0) = y_0$, there exists a unique solution if:

1. Continuity: $f(x, y)$ is continuous in a neighborhood around $(x_0, y_0)$.
2. Lipschitz condition: $f(x, y)$ satisfies a Lipschitz condition with respect to $y$, meaning there is a constant $L$ such that for all $y_1$ and $y_2$ in the neighborhood: $|f(x, y_1) - f(x, y_2)| \leq L |y_1 - y_2|.$

Now, let's apply these conditions to the given problem.

Step 1: Continuity of $f(y) = |y|^{3/4}$

The function $f(y) = |y|^{3/4}$ is continuous for all values of $y$, including at $y = 0$. Therefore, the continuity condition is satisfied.

Step 2: Lipschitz Condition

We now check whether $f(y) = |y|^{3/4}$ satisfies a Lipschitz condition near $y = 0$.

We compute the derivative of $f(y) = |y|^{3/4}$ to examine its behavior:

• For $y > 0$, $f'(y) = \frac{3}{4} y^{-1/4}$.
• For $y = 0$, the derivative becomes problematic because $y^{-1/4} \to \infty$ as $y \to 0^+$.

Thus, $f(y) = |y|^{3/4}$ does not satisfy a Lipschitz condition at $y = 0$, because the derivative is not bounded near $y = 0$.

Conclusion

Since the function $f(y) = |y|^{3/4}$ does not satisfy the Lipschitz condition near $y = 0$, the Picard-Lindelöf theorem does not guarantee uniqueness of the solution at $y(0) = 0$. However, existence is still assured due to the continuity of $f(y)$.

Therefore, the solution exists, but it may not be unique.

Would you like further details or clarification on any part of the explanation?

Here are 5 questions to explore the topic further:

1. What is the general solution to the differential equation $\frac{dy}{dx} = |y|^{3/4}$?
2. How would the behavior of the solution change if $y(0) \neq 0$?
3. Can the Picard-Lindelöf theorem be extended to higher-order differential equations?
4. What happens if the exponent in $|y|^{3/4}$ is changed? Does uniqueness always fail for such exponents?
5. How can numerical methods be used to approximate solutions in cases where uniqueness fails?

Tip: For differential equations where uniqueness might fail, studying the direction fields or using numerical methods like Euler's method can provide insight into the solution's behavior.