The image you've uploaded presents an example discussing initial value problems (IVPs) in differential equations, particularly demonstrating that an IVP can have multiple solutions. Let's break down the key components and the related problem:

Problem Description:

The differential equation given is: $\frac{dy}{dx} = xy^{1/2}, \quad y(0) = 0$ This represents an initial value problem (IVP) where the goal is to determine the solution $y(x)$ that satisfies both the differential equation and the initial condition $y(0) = 0$.

Analysis:

The text explains that this IVP has at least two solutions:

1. $y(x) = 0$ (the trivial solution)
2. $y(x) = \frac{x^2}{4}$

Both solutions satisfy the differential equation and the initial condition.

Why Multiple Solutions?

The differential equation $\frac{dy}{dx} = xy^{1/2}$ is not "Lipschitz continuous" at $y = 0$, which means that the conditions for the existence and uniqueness theorem of differential equations are not fully satisfied. This leads to the possibility of multiple solutions passing through the point $(0, 0)$, as illustrated in the graph.

Existence of Two Solutions:

• $y = 0$: If $y(x) = 0$, the derivative $\frac{dy}{dx} = 0$ satisfies the equation, so this is a valid solution.
• $y = \frac{x^2}{4}$: If $y(x) = \frac{x^2}{4}$, then: $\frac{dy}{dx} = \frac{2x}{4} = \frac{x}{2}, \quad y^{1/2} = \left( \frac{x^2}{4} \right)^{1/2} = \frac{|x|}{2}$ which also satisfies the differential equation $\frac{dy}{dx} = xy^{1/2}$.

Conclusion:

Thus, the IVP has at least two solutions: $y(x) = 0$ and $y(x) = \frac{x^2}{4}$.

Would you like a detailed explanation of the Existence and Uniqueness Theorem or how these solutions are derived in more depth?

Related Questions:

1. What conditions ensure the uniqueness of a solution to an IVP?
2. How does Lipschitz continuity affect the solution to a differential equation?
3. Could this IVP have more than two solutions? Why or why not?
4. How do you verify if a given function is a solution to a differential equation?
5. What is the geometric interpretation of multiple solutions for the same IVP?

Tip:

In solving differential equations, always check the conditions of the Existence and Uniqueness Theorem to determine if the solution is guaranteed to be unique.