Problem 20 - Advanced Engineering Mathematics

Question

Determine a region of the \(x y\) -plane for which the given differential equation would have a unique solution whose graph passes through a point \(\left(x_{0}, y_{0}\right)\) in the region. $$ \frac{d y}{d x}-y=x $$

Step-by-Step Solution

Answer

The region is the entire \(xy\)-plane.

Step 1: Identify the Form of Differential Equation

The given differential equation is \( \frac{dy}{dx} - y = x \). We can rewrite it as \( \frac{dy}{dx} = y + x \). This is a first-order linear differential equation which can be written in the form \( \frac{dy}{dx} = f(x, y) \), where \( f(x, y) = y + x \).

Step 2: Find Partial Derivatives

The existence of a unique solution through a point \((x_0, y_0)\) depends on the function \(f(x, y)\) and its partial derivative with respect to \(y\). Calculate the partial derivative with respect to \(y\):\[ \frac{\partial f}{\partial y} = \frac{\partial (y + x)}{\partial y} = 1. \]

Step 3: Apply the Existence and Uniqueness Theorem

According to the Existence and Uniqueness Theorem (also known as the Picard-Lindelöf theorem), if the functions \(f(x, y)\) and \(\frac{\partial f}{\partial y}\) are continuous in a region \(R\) containing the point \((x_0, y_0)\), then there exists a unique solution to the differential equation in that region. Since both \(y + x\) and its derivative with respect to \(y\) (which is \(1\)) are continuous everywhere, the function is continuous in all of \(\mathbb{R}^2\).

Step 4: Determine Region for Unique Solution

Since both \(f(x, y) = x + y\) and its partial derivative \(\frac{\partial f}{\partial y} = 1\) are continuous everywhere in the \(xy\)-plane, the region where the differential equation has a unique solution is the entire \(xy\)-plane.

Key Concepts

First-order Linear Differential Equations

A first-order linear differential equation involves derivatives of a function and its variables, specifically in the form \( \frac{dy}{dx} + P(x)y = Q(x) \). In our case, the differential equation \( \frac{dy}{dx} = y + x \) falls into this category.

Here are its key characteristics:

• "First-order" means the equation involves the first derivative of \( y \) with respect to \( x \).
• "Linear" signifies that \( y \) and \( \frac{dy}{dx} \) are each to the first power and not multiplied together.

Rewriting the equation in the standard linear form, we have \( \frac{dy}{dx} - y = x \). Such equations, when recognized properly, can be solved using specific strategies such as the integrating factor, making it possible to find solutions that describe how \( y \) changes with \( x \). Understanding these equations is crucial because they often model numerous real-world phenomena ranging from population growth to electrical circuits.

Partial Derivatives

Partial derivatives are an essential concept when dealing with functions of multiple variables, like our function \( f(x, y) = y + x \). The idea behind a partial derivative is to specify how a function changes as one of its input variables changes, holding the others constant.

In the exercise, we calculate the partial derivative of \( f(x, y) \) with respect to \( y \), which involves:

• Treating \( x \) as a constant.
• Determining how \( f(x, y) \) changes with small changes in \( y \).

This yields \( \frac{\partial f}{\partial y} = 1 \), indicating that for every unit increase in \( y \), the function increases by the same amount, unaffected by \( x \).

Partial derivatives are pivotal in analyzing the behavior of multi-variable functions, especially when determining things like slopes in various directions, which contribute to unique solution conditions in differential equations.

Picard-Lindelöf Theorem

The Picard-Lindelöf theorem is a central pillar in determining the existence and uniqueness of solutions to differential equations.

This theorem posits:

• If \( f(x, y) \) and its partial derivative with respect to \( y \) are continuous in a region \( R \), then there is a unique solution to the differential equation through any point \((x_0, y_0)\) in \( R \).

In our case, both \( f(x, y) = y + x \) and its partial derivative \( \frac{\partial f}{\partial y} = 1 \) are continuous across the entire \( xy \)-plane.

This universality means any point \((x_0, y_0)\) in this plane will have a unique solution curve passing through it. The theorem thus guarantees that, within this limitless region, the behavior of solutions is predictable and can be trusted to be consistent, a valuable assurance for mathematicians and scientists alike.