Question

Solve the initial-value problems. $$ \frac{d y}{d x}=-y^{2}+x y+1 ; \text { given solution } f(x)=x $$

Short answer

The solution for the initial-value problem is \(y(x) = x\), as it has been verified that the given solution satisfies the given differential equation.

Step-by-step solution

Incorporate the given solution into the differential equation

We're given the solution f(x) = x. To see if it satisfies the differential equation, substitute this function into the equation: $$ \frac{d}{dx}(x) = -x^2 + x(x) + 1 $$

Differentiate the given function

Now we differentiate the given function f(x) with respect to x: $$ \frac{d}{dx}(x) = 1 $$

Verify if the given solution satisfies the differential equation

Now we substitute the result of Step 2 back into the differential equation and check if it satisfies the equation: $$ 1 = -x^2 + x^2 + 1 $$ Simplifying, we have: $$ 1 = 1 $$ This confirms that the given solution f(x) = x indeed satisfies the given differential equation.

State the solution of the initial-value problem

Since we have verified that the given solution satisfies the differential equation, we can now state the solution of the initial-value problem: The solution for the initial-value problem is \(y(x) = x\).

Key concepts

Differential Equation

A differential equation is a mathematical equation that relates a function with its derivatives. In the context of this exercise, we have a first-order differential equation, which means it involves the first derivative of the function. The equation provided is \( \frac{d y}{d x} = -y^2 + x y + 1 \). This equation describes how the derivative of \( y \) with respect to \( x \) changes based on the values of \( y \) and \( x \) themselves. Differential equations are core to modeling various physical, biological, and engineering phenomena. They can describe how quantities change over time or space and are used in fields from physics to economics. Understanding how to solve them can provide insights into complex dynamic systems.In the initial-value problem, the goal is to find a function \( y(x) \) that satisfies the differential equation given initial conditions.

Solution Verification

Solution verification involves checking whether a proposed solution fulfills the requirements of the original problem, in this case, the differential equation. We are given \( f(x) = x \) as a proposed solution, and we need to verify its validity by substituting it back into the differential equation. To perform verification:

• First, substitute \( f(x) = x \) into both sides of the equation.
• Compute the derivative of the solution, in this case \( \frac{d}{dx}(x) = 1 \).
• Plug the result into the left-hand side of the equation to see if both sides match.

When both sides of the equation equal each other, it confirms the validity of the solution. In our exercise, substituting values shows that \( 1 = 1 \), indicating \( f(x) = x \) is indeed a valid solution.

Derivatives

Derivatives represent the rate at which a function is changing at any given point, which is crucial for solving differential equations. The derivative of a function \( f(x) \) is denoted as \( \frac{d}{dx}f(x) \). In this exercise, the derivative of the function \( f(x) = x \) with respect to \( x \) is simply 1.Derivatives are significant in understanding growth patterns, trends, and rates of change in various real-world contexts:

• In physics, they describe velocity and acceleration.
• In economics, they can forecast changes in profit or cost functions.
• In biology, they may represent population growth rates.

By computing the derivative and using it in the differential equation, we ensure that the proposed solution adheres consistently to the described dynamic behavior.

Function Substitution

Function substitution is an essential technique when dealing with differential equations. In this exercise, we use it to substitute \( f(x) = x \) into the differential equation to check if it solves it. The steps involve:

• Replacing \( y \) in the equation with \( x \) (since \( f(x) = x \)).
• Substituting \( \frac{d}{dx}(x) = 1 \) into the left side of the differential equation.
• Simplifying the equation to verify the truth of the equality.

Function substitution helps simplify complex expressions and verify candidate solutions against the structure of the differential equation. This concept underlines how flexible and powerful substitution is in checking solutions and serves as a foundational method in calculus.