Question

Let \(f(x, y)\) be a continuous bounded function on an open set \(\Omega\) in \(\mathbf{R}^{2}\) that is uniformly Lipschitz in \(y\) on \(\Omega\) (there is a constant \(K\) such that \(\left|f\left(x, y_{1}\right)-f\left(x, y_{2}\right)\right| \leq K\left|y_{1}-y_{2}\right|\) for every \(\left.\left(x, y_{1}\right),\left(x, y_{2}\right) \in \Omega\right)\) Let \(\left(x_{0}, y_{0}\right) \in \Omega\). Show that there is \(\delta>0\) such that on \(\left[x_{0}-\delta, x_{0}+\delta\right]\) there is a unique continuously differentiable solution to \(\frac{d y}{d x}=f(x, y)\) with the initial condition \(y\left(x_{0}\right)=y_{0}\)

Short answer

By the Picard-Lindelöf theorem, the function \( f(x, y) \) guarantees a unique solution for \( \frac{dy}{dx} = f(x, y) \) on \[x_{0} \pm \delta\], for some \( \delta > 0 \).

Step-by-step solution

State the Problem and Given Conditions

We are given a function \( f(x, y) \) that is continuous, bounded, and uniformly Lipschitz in \( y \) on an open set \( \Omega \) in \( \mathbf{R}^{2} \). Specifically, there is a constant \( K \) such that \( |f(x, y_{1}) - f(x, y_{2})| \leq K|y_{1}-y_{2}| \) for all pairs \((x, y_{1}), (x, y_{2}) \in \Omega \). Our goal is to show that there is a \(\delta>0\) such that on \( [x_{0}-\delta, x_{0}+\delta] \), there is a unique, continuously differentiable solution to the initial value problem (IVP) \(\frac{dy}{dx}=f(x,y) \) with \( y(x_{0})=y_{0} \).

Local Existence and Uniqueness Theorem

According to the Picard-Lindelöf theorem (also known as the Cauchy-Lipschitz theorem), if a function \( f(x, y) \) is continuous in \( x \) and Lipschitz continuous in \( y \) on some rectangle containing the initial point \( (x_{0}, y_{0}) \), then there exists \(\delta>0\) such that a unique continuously differentiable solution to the IVP exists on the interval \( [x_{0}-\delta, x_{0}+\delta]. \)

Verify Lipschitz Condition

Given that \( f(x,y) \) is uniformly Lipschitz in \( y \) on \( \Omega \), for any \( y_{1}, y_{2} \in \Omega \), \left| f(x, y_{1}) - f(x, y_{2}) \right| \leq K\| y_{1} - y_{2} \|. This confirms the Lipschitz condition required by the Picard-Lindelöf theorem.

Conclusion from the Theorem

By the Picard-Lindelöf theorem, the function \( f(x, y) \) being Lipschitz in \( y \) guarantees that there is an interval \( [x_{0}-\delta, x_{0}+\delta] \), for some \(\delta>0\), where a unique continuously differentiable solution exists for the IVP given by \( \frac{dy}{dx} = f(x, y) \) with \( y(x_{0}) = y_{0} \).

Key concepts

Lipschitz Condition

The Lipschitz condition is a crucial part of understanding the solutions to certain differential equations. In our problem, the function \( f(x, y) \) needs to satisfy this condition to apply the Picard-Lindelöf theorem. But what exactly is the Lipschitz condition?
This condition ensures that the change in the function \( f \) does not vary too rapidly with respect to changes in \( y \). More formally, a function \( f(x, y) \) is said to be Lipschitz continuous in \( y \) on an open set \( \Omega \) if there exists a constant \( K \) such that for all \( (x, y_1), (x, y_2) \) in \( \Omega\), the following inequality holds:
\[|f(x, y_1) - f(x, y_2)| \leq K|y_1 - y_2|\]This inequality means that the function \( f \) doesn’t change too wildly; its rate of change is bounded by a constant factor of the difference in \( y \).
Understanding this concept helps us see why \( f \) being Lipschitz continuous ensures the possibility of having a unique solution to our initial value problem (IVP). The bounded behavior avoids runaway solutions, making our differential equation more 'well-behaved.'

Initial Value Problem

An initial value problem (IVP) is a type of differential equation accompanied by specified values at a starting point. In our exercise, the problem is defined by the differential equation:\[\frac{dy}{dx} = f(x, y)\]and the initial condition:\[y(x_0) = y_0\]These conditions essentially tell us the behavior of the function \( y \) at an initial point \( x_0 \). They are key to finding a specific solution to the differential equation, out of many possible ones, which passes through the point \( (x_0, y_0) \).
When we talk about solving an IVP, we are looking for a function \( y(x) \) that satisfies both the differential equation and the initial condition. The goal is to find such a function that is well-defined and behaves consistently over some interval around \( x_0 \).
This is where the Picard-Lindelöf theorem steps in, providing a foundation for proving the existence and uniqueness of the solution under certain conditions, such as the Lipschitz condition. It guarantees us that, given our initial conditions, there’s a unique solution that won’t 'split' into multiple different paths as we move away from \( x_0 \).

Unique Solution

The concept of a unique solution is central to many mathematical problems, especially in the context of differential equations like our initial value problem (IVP). In simpler terms, a unique solution means that, given the initial conditions, there is exactly one function that satisfies both the differential equation and the initial value.
In our exercise, we need to show that there is a \( \delta > 0 \) such that on the interval \([x_0 - \delta, x_0 + \delta] \), the differential equation:\[\frac{dy}{dx} = f(x, y)\]with the initial condition:\[y(x_0) = y_0\]has precisely one solution that is continuously differentiable.
The Picard-Lindelöf theorem confirms this by guaranteeing the existence of such a unique solution if the function \( f(x, y) \) meets the required conditions, such as being Lipschitz continuous. Without uniqueness, you'd have multiple solutions for the same initial condition, leading to ambiguity and making the model unreliable.
This uniqueness is important in ensuring that the differential equation models real-world phenomena accurately and predictably. For example, in physics and engineering, where the behavior of systems is continually modeled using differential equations, having a unique solution is critical for accurate predictions.