Question

Using an Integrating Factor The expression \(u(x)\) is an integrating factor for \(y^{\prime}+P(x) y=Q(x)\) . Which of the following is equal to \(u^{\prime}(x) ?\) Verify your answer. $$ \begin{array}{ll}{\text { (a) } P(x) u(x)} & {\text { (b) } P^{\prime}(x) u(x)} \\ {\text { (c) } Q(x) u(x)} & {\text { (d) } Q^{\prime}(x) u(x)}\end{array} $$

Short answer

The derivative of the integrating factor \(u^{\prime}(x) = P(x) u(x)\), so the correct answer is (a) \(P(x) u(x)\).

Step-by-step solution

Understanding Integrating Factors

First, understand that an integrating factor is a function that is used to solve linear differential equations of the form \(y^{\prime}+P(x) y=Q(x)\). The integrating factor, in this case, is \(u(x)\), and the differential equation has a coefficient \(P(x)\) and a function \(Q(x)\).

Understand the Derivative of the Integrating Factor

The derivative of the integrating factor \(u(x)\), or \(u^{\prime}(x)\), is defined as \(u^{\prime}(x) = u(x)P(x)\). This is based on the property of integrating factors.

Matching the Answer

Now you match the definition of \(u^{\prime}(x)\) from step 2, which is \(u(x)P(x)\) with the options given in the exercise, and find the answer is (a) \(P(x) u(x)\).

Key concepts

Linear Differential Equations

Linear differential equations are a fundamental part of mathematics, used to model many real-world phenomena. They typically have the form:

• First-order: \( y^{\prime} + P(x)y = Q(x) \)
• Higher-order: Involving higher derivatives, but can often be simplified into first-order forms.

In the equation format above, \( y^{\prime} \) represents the derivative of \( y \) with respect to the variable \( x \), \( P(x) \) is a coefficient function, and \( Q(x) \) is a driving function or source term. Applying an integrating factor is a standard method for solving such equations.
By choosing an appropriate integrating factor, you can transform a complex differential equation into one that is simpler to solve, often reducing it to a standard integration problem.

Derivative of a Function

The derivative of a function often measures how a function's output values change as its input changes. In calculus, it provides critical insights into the behavior of functions, especially when analyzing curves and slopes. For example, if you have a function \( f(x) \), its derivative \( f^{\prime}(x) \) tells you the rate of change of \( f \) with respect to \( x \).
Derivatives can be used to find:

• Slopes of tangents to curves at given points.
• Instantaneous rates of change, like velocity and acceleration in physics.
• Critical points where functions reach maximum or minimum values.

When solving linear differential equations, understanding the derivative helps because the process often involves differentiating unknown and known functions. For integrating factors, knowing the derivative formula is key for determining the correct factor to simplify the equation.

Function of a Single Variable

Functions of a single variable are crucial because they relate each value from a domain (input) to a value from a range (output). Consider a function \( f(x) \). Here, \( x \) is the independent variable, and the function produces an output based on \( x \) alone.
These functions provide a foundational framework for understanding broader mathematical concepts, serving as building blocks for more complex multi-variable functions.
In the context of differential equations:

• The functions \( P(x) \) and \( Q(x) \) are examples of functions of a single variable, where the behavior of these functions can directly influence the outcome of the equation.
• Simplifying and manipulating these functions of a single variable are often necessary to solve or approximate solutions to equations.

Having a solid grasp of single-variable functions is essential for navigating topics such as derivatives and linear differential equations, as they frequently interact in solving mathematical problems.