Problem 37 - Calculus

Question

Decide whether the statement is true or false. Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=2 x-y .\) Justify your answer. $$f^{\prime}(x)=2 x-f(x)$$

Step-by-Step Solution

Answer

True, based on matching the differential equation.

Step 1: Understanding the Given Equation

We are given that the function \( y = f(x) \) is a solution to the differential equation \( \frac{dy}{dx} = 2x - y \). This initially implies that the derivative of \( y \) with respect to \( x \), denoted \( f'(x) \), equals \( 2x - f(x) \), as this directly translates the expression given.

Step 2: Restating the Equation

Restate the given differential equation in terms of the function notation: \( f'(x) = 2x - f(x) \). This is exactly the statement we are asked to verify: "\( f'(x) = 2x - f(x) \)."

Step 3: Analyzing the Statement

Since the problem statement asks if \( f'(x) = 2x - f(x) \) is true given \( y = f(x) \) solves \( \frac{dy}{dx} = 2x - y \), we compare both. The equation \( \frac{dy}{dx} = 2x - y \) directly translates to \( f'(x) = 2x - f(x) \), confirming the given statement.

Step 4: Conclusion

Since the translated expression \( f'(x) = 2x - f(x) \) matches the problem's differential equation, we conclude that the statement "\( f'(x) = 2x - f(x) \)" is true.

Key Concepts

Solution Verification

Solution verification is a crucial part of solving differential equations. It helps confirm whether our proposed solution indeed satisfies the initial equation. In this exercise, we are tasked with verifying the statement \( f'(x) = 2x - f(x) \). This statement serves as the derivative of the function \( y = f(x) \), which is claimed to satisfy the differential equation \( \frac{dy}{dx} = 2x - y \). To verify a solution, follow these steps:

• Substitute the proposed solution into the differential equation.
• Check if both sides of the equation are equal after simplification.

By substituting \( f'(x) \) with \( 2x - f(x) \) and comparing each side in this exercise, we find they match perfectly. This confirms that the given function indeed solves the equation.

First-Order Differential Equations

A first-order differential equation involves derivatives of the first degree and no higher derivatives. A general form is written as \( \frac{dy}{dx} = g(x, y) \), where \( g(x, y) \) is some function of \( x \) and \( y \). In our case, the differential equation \( \frac{dy}{dx} = 2x - y \) is a classic example of a linear first-order differential equation. Key Characteristics of First-Order Differential Equations:

• They only involve first derivatives, meaning only \( \frac{dy}{dx} \), and not involvement of any \( \frac{d^2y}{dx^2} \) or higher derivatives.
• The equation describes the rate at which \( y \) changes concerning \( x \).
• Such equations are pivotal in modeling several real-world systems, including population growth and decay processes.

For the equation \( \frac{dy}{dx} = 2x - y \), finding solutions typically involves techniques like separation of variables or integrating factors, though sometimes direct substitution verifies a proposed solution more efficiently.

Function Notation

Function notation is an essential tool that describes the rules used for transforming inputs (independent variables) into outputs (dependent variables). In calculus and differential equations, function notation simplifies expressions and helps communicate mathematical ideas clearly. Key Points:

• Function notation is represented as \( y = f(x) \), where \( y \) is the output and \( x \) is the input.
• It provides a concise way to compare equations, like the translation of a differential equation into \( f'(x) = 2x - f(x) \).

Here, \( f'(x) \) represents the derivative of \( f \), describing how \( y = f(x) \) changes. By using function notation, we directly relate the function's behavior through its derivatives, crucial in verifying solutions for differential equations.