Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-14, solve each differential equation. $$ \frac{d y}{d x}-\frac{y}{x}=x e^{x} $$

Step-by-Step Solution

Answer

The solution is \( y = x e^x + Cx \).

Step 1: Identify type of differential equation

The given differential equation is \( \frac{dy}{dx} - \frac{y}{x} = x e^{x} \). This is a first-order linear differential equation of the form \( \frac{dy}{dx} + P(x) y = Q(x) \). Here, \( P(x) = -\frac{1}{x} \) and \( Q(x) = x e^{x} \).

Step 2: Find the integrating factor

The integrating factor \( \mu(x) \) is found using the formula \( \mu(x) = e^{\int P(x) \, dx} \). In this case, \( P(x) = -\frac{1}{x} \), so we have:\[ \mu(x) = e^{\int -\frac{1}{x} \, dx} = e^{-\ln|x|} = \frac{1}{x} \].

Step 3: Multiply the equation by the integrating factor

Multiply the entire differential equation by the integrating factor \( \frac{1}{x} \):\[ \frac{1}{x} \frac{dy}{dx} - \frac{1}{x^2} y = e^{x} \].This makes the left-hand side a derivative of a product.

Step 4: Rewrite as a derivative of a product

Notice that the left-hand side can be rewritten as a single derivative:\[ \frac{d}{dx} \left( \frac{y}{x} \right) = e^x \].

Step 5: Integrate both sides

Integrate both sides with respect to \( x \):\[ \int \frac{d}{dx} \left( \frac{y}{x} \right) \, dx = \int e^x \, dx \].This gives \( \frac{y}{x} = e^x + C \), where \( C \) is a constant of integration.

Step 6: Solve for \( y \, \)

Multiply through by \( x \) to isolate \( y \):\[ y = x e^x + Cx \].

Step 7: Final Answer

The general solution to the differential equation is:\[ y = x e^x + Cx \], where \( C \) is an arbitrary constant.

Key Concepts

Integrating Factor

The concept of an integrating factor is a key tool in solving first-order linear differential equations. An integrating factor helps restructure the differential equation such that it can be solved more easily by integration.
To find the integrating factor, we examine the given form of the differential equation: \( \frac{dy}{dx} + P(x)y = Q(x) \). The formula used for the integrating factor \( \mu(x) \) is \( e^{\int P(x) \, dx} \). This method is essential because it converts the equation into an exact differential, which is much easier to integrate.
For the equation \( \frac{dy}{dx} - \frac{y}{x} = x e^{x} \), \( P(x) = -\frac{1}{x} \). We compute the integrating factor as \( \mu(x) = e^{-\ln|x|} = \frac{1}{x} \). Applying this transformation is what allows the left-hand side to be written as a derivative of a product in the subsequent steps.

Constant of Integration

When solving differential equations, the constant of integration \( C \) emerges naturally from the integration process. This constant represents the indefinite nature of antiderivatives, indicating that there are infinitely many solutions to the differential equation, each differing by a constant.
After integrating the transformed equation, we have \( \frac{y}{x} = e^x + C \), where \( C \) is added as the constant of integration. This constant can be determined if we have an initial condition or a specific point the solution must satisfy. Without additional information, \( C \) remains arbitrary, reflecting the general family of solutions to the problem.
Understanding the role of \( C \) is crucial because it emphasizes the adaptable nature of solutions in a linear differential equation context.

General Solution

The general solution of a differential equation not only solves the equation, but it also encompasses all possible particular solutions. This solution includes the arbitrary constant, which accounts for the diversity of situations the equation might model.
For the given equation, after integration and manipulation, our general solution is \( y = x e^x + Cx \). This expression represents the entire family of functions that satisfy the original differential equation.
The term "general solution" implies that, by substituting different values for \( C \), we can find specific solutions that fit any given initial condition or boundary situation. This flexibility is particularly potent in applications where specific physical scenarios determine \( C \).

Solving Differential Equations Step by Step

Solving differential equations step by step involves a structured approach to reach the final solution. This method ensures not only correctness but also clarity in understanding how each step logically proceeds to the next.
1. **Identify the Equation Type**: - Recognize the form and elements of the equation.2. **Compute Integrating Factor**: - Apply the formula \( \mu(x) = e^{\int P(x) \, dx} \) to find an integrating factor.3. **Multiply by the Integrating Factor**: - Use the integrating factor to transform the differential equation.4. **Rewrite as a Derivative of a Product**: - Simplify the equation by expressing it as the derivative of a product of functions.5. **Integrate Both Sides**: - Integrate to find the general solutions, introducing a constant of integration.6. **Solve for Dependent Variable**: - Rearrange the equation to express the dependent variable explicitly.
These carefully crafted steps offer a roadmap from the initial problem to the general solution, making the process of solving differential equations manageable and logical.