Problem 110 - Thomas Calculus

Question

\(\begin{array}{l}{\text { Uniqueness of solutions } \text { If differentiable functions } y=F(x)} \\ {\text { and } y=g(x) \text { both solve the initial value problem }}\end{array}\) $$\frac{d y}{d x}=f(x), \quad y\left(x_{0}\right)=y_{0}$$ \(\begin{array}{l}{\text { on an interval } I, \text { must } F(x)=G(x) \text { for every } x \text { in } R \text { ? Give reasons }} \\ {\text { for your answer. }}\end{array}\)

Step-by-Step Solution

Answer

Yes, if the conditions for uniqueness are met, then \( F(x) = g(x) \) for all \( x \) in the interval \( I \).

Step 1: Define the Initial Value Problem

The initial value problem involves a differential equation \( \frac{dy}{dx} = f(x) \) with an initial condition \( y(x_0) = y_0 \). It asks whether two differentiable functions \( y = F(x) \) and \( y = g(x) \) that solve this problem are necessarily equal over the domain \( I \).

Step 2: State the Condition for Uniqueness

The uniqueness of solutions to the initial value problem, as defined, requires the function \( f(x) \) to satisfy conditions such as the Lipschitz condition on the interval \( I \). By the uniqueness theorem, if \( f(x) \) is continuous and satisfies a Lipschitz condition, then the differential equation has a unique solution passing through the point \( (x_0, y_0) \).

Step 3: Analyze the Given Functions

Since both \( y = F(x) \) and \( y = g(x) \) solve the differential equation and adhere to the same initial condition \( y(x_0) = y_0 \), the conditions for uniqueness imply that \( F(x) \) and \( g(x) \) are indeed the same function for every \( x \) in \( I \).

Step 4: Conclude Based on Uniqueness Theorem

Due to the uniqueness theorem, which states that solutions that satisfy the initial condition in the presence of a Lipschitz condition on \( f(x) \) are unique, we conclude that \( F(x) \) must equal \( g(x) \) for all \( x \) in the interval \( I \).

Key Concepts

Initial Value Problem

An initial value problem (IVP) in mathematics is a problem where you are given a differential equation, along with a specific value at the starting point. The general form is:

• Differential Equation: \( \frac{dy}{dx} = f(x) \)
• Initial Condition: \( y(x_0) = y_0 \)

The initial condition specifies the value of the function at a particular point, \( x_0 \), thereby defining the starting point of the solution. This is crucial when determining the behavior of the function over the interval. Understanding the initial value is essential in demonstrating the uniqueness of the solution, as any valid function must pass through the point \((x_0, y_0)\). This lays the groundwork for further analyses and application of uniqueness theorems.

Differential Equation

Differential equations are equations that involve the derivatives of a function. The derivative, \( \frac{dy}{dx} \), provides information about how a function changes at any given point. Commonly found in physics, engineering, and other disciplines, differential equations model phenomena that involve rates of change.

In the context of the initial value problem, the differential equation presents a guide to how the function should behave. Once paired with an initial condition, the path of the function is dictated by solving the differential equation, which acts as a set of instructions. The differential equation here is \( \frac{dy}{dx} = f(x) \), where solving it gives us a function, \( y(x) \), that satisfies this rate of change over a domain or interval \( I \). Hence, the differential equation governs the dynamic aspects of how a solution evolves from the initial state.

Lipschitz Condition

The Lipschitz condition is a mathematical condition used to ensure the uniqueness of solutions in differential equations. It refers to a particular set of bounds on the function \( f(x) \) or its derivatives.

• The condition implies that the function does not change too rapidly, and a small change in \( x \) results in a proportionally small change in \( f(x) \).
• If \( f(x) \) satisfies a Lipschitz condition on the interval \( I \), then the differential equation \( \frac{dy}{dx} = f(x) \) admits a unique solution.

This condition is vital since it prevents oscillatory behaviors or extreme changes in \( y \) that would otherwise allow multiple functions to satisfy the same initial value problem. Thus, when \( f(x) \) adheres to the Lipschitz condition, it provides assurance of the solution's uniqueness, confirming that two functions solving the IVP cannot diverge while still satisfying the prescribed conditions.

Differentiable Functions

Differentiable functions are functions that possess a derivative at every point within their domain. These functions are smooth and continuously differentiable, which implies that there are no jumps, corners, or cusps in the graph of the function.

For the discussion of the uniqueness theorem, considering functions \( y = F(x) \) and \( y = g(x) \) that solve a differential equation is important. Both must be differentiable. Differentiability ensures that the functions are adequately smooth to be compared pointwise and that their derivatives correspond to the differential equation.

• Within the context of the IVP, this requirement guarantees that the mathematical operations performed through the solution process are valid.
• Diffentiable nature of \( F(x) \) and \( g(x) \) allows them not only to match the initial condition but also to match continually as they evolve from \( x_0 \) throughout the entire interval \( I \).

This attribute of differentiability supports the uniqueness theorem by ensuring the consistency and reliability of the solution path.