Problem 38 - 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. There could be more than one value of \(x\) such that \(f^{\prime}(x)=1\) and \(f(x)=5\)

Step-by-Step Solution

Answer

The statement is false; there is only one value of \(x = 3\) satisfying both conditions.

Step 1: Understand the Given Differential Equation

We are given the differential equation \( \frac{dy}{dx} = 2x - y \). If \( y = f(x) \) satisfies this equation, then for each point where \( x = a \), \( \frac{dy}{dx} \) at \( x = a \) would be \( f^\prime(a) = 2a - f(a) \). This is the condition given within the statement.

Step 2: Requirement Analysis

The question asks if there can be more than one value of \( x \) such that \( f^\prime(x) = 1 \) and \( f(x) = 5 \). So, we need to examine if there can be multiple values of \( x \) that satisfy both conditions.

Step 3: Solve for x Given f'(x) = 1

Substitute \( f'(x) = 1 \) in the differential equation: \( 1 = 2x - f(x) \). Hence, \( 2x - 5 = 1 \), since \( f(x) = 5 \). This simplifies to \( 2x = 6 \) which gives \( x = 3 \).

Step 4: Check Condition for Multiple Values of x

Since solving the equations \( f'(x) = 1 \) and \( f(x) = 5 \) gives a unique solution \( x = 3 \), there are not multiple values of \( x \) meeting both conditions. Thus, there cannot be more than one \( x \) satisfying both conditions simultaneously.

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus. They represent the rate at which a quantity changes with respect to another variable. Think of it like how fast a car's speedometer reading changes over time. In our exercise, the derivative \( \frac{dy}{dx} = 2x - y \) describes how the function \( y = f(x) \) changes its slope or gradient at any point \( x \). The point where \( f'(x) = 1 \) indicates that the slope of the tangent to the curve at the corresponding point is 1.
The derivative can tell us a lot about the behavior of a function. It can show whether the function is increasing or decreasing at a given point. Also, understanding the derivative is crucial for finding solutions to differential equations. With \( f'(x) = 1 \), it helps us to specifically figure out the values that make the slope equal to 1 in the context of our problem.

Solutions to Differential Equations

A differential equation relates a function with its derivatives. It might sound complex, but it's a way to establish how functions behave in various situations. In our exercise, the equation \( \frac{dy}{dx} = 2x - y \) helps us understand the dynamics between \( y \) and \( x \).
Finding a solution like \( y = f(x) \) means identifying a function that satisfies this relation for all relevant values of \( x \). To solve the differential equation in this problem, we ended up using specific conditions \( f'(x) = 1 \) and \( f(x) = 5 \). By substituting these conditions into the equation, we solved for \( x \) and found \( x = 3 \) as the solution. This type of problem demonstrates why understanding differential equations is key in many scientific and engineering problems, as it allows us to model and predict real-world behaviors.

Existence and Uniqueness of Solutions

The existence and uniqueness of solutions in the context of differential equations refer to whether a solution actually exists for a given differential system and whether that solution is uniquely determined. In simpler terms, it asks if there's at least one solution to the equation and if there is, is it the only one?
For the differential equation given in our exercise, the condition \( f'(x) = 1 \) and \( f(x) = 5 \) lead to the singular solution \( x = 3 \). This suggests not only the existence of a solution but also that it is unique under the specified conditions. Such results are often guaranteed by theorems like the Picard-Lindelöf theorem, which tells us that under certain circumstances, there's exactly one solution passing through a given point.
Understanding these concepts helps us interpret real-life systems where such properties ensure predictability and reliability of models based on differential equations.