Final answer:
A first-order ODE example with a non-unique solution at a given initial condition is dy/dx = sqrt(|y|), which for y(0) = 0 has multiple solutions such as y(x) = 0 and y(x) = x^2/4. The ODE does not satisfy the Lipschitz condition, allowing for non-unique solutions.
Step-by-step explanation:
An example of a first-order Ordinary Differential Equation (ODE) that has a non-unique solution at a given initial condition y(x_0) = y_0 is best represented by the function which does not satisfy the Lipschitz condition near the initial value. For instance, consider the following ODE:
dy/dx = sqrt(|y|)
For the initial condition y(0) = 0, this differential equation has multiple solutions, including y(x) = 0 and y(x) = x^2/4 for x ≥ 0. When you sketch a graph of these functions, it's evident that different solutions pass through the initial condition (x_0, y_0), demonstrating the non-uniqueness of the solution.
To further explain, in a region around the point (0, 0), the function sqrt(|y|) is not Lipschitz continuous, and thus the Picard-Lindelöf theorem on the existence and uniqueness of solutions to ODEs does not apply here, allowing for multiple solutions emanating from the same initial point.