Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Solve each differential equation. $$ \frac{d y}{d x}+2 y=x $$

Step-by-Step Solution

Answer

The solution to the differential equation is \(y = \frac{1}{2}x - \frac{1}{4} + Ce^{-2x}\).

Step 1: Identify the type of differential equation

This differential equation is a first-order linear differential equation as it can be written in the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\), with \(P(x) = 2\) and \(Q(x) = x\).

Step 2: Compute the integrating factor

The integrating factor \(\mu(x)\) is found using \(\mu(x) = e^{\int P(x) \, dx}\). Since \(P(x) = 2\), we have \(\mu(x) = e^{\int 2 \, dx} = e^{2x}\).

Step 3: Multiply the entire equation by the integrating factor

Multiply every term in the differential equation by \(e^{2x}\): \[e^{2x} \frac{dy}{dx} + 2e^{2x}y = xe^{2x}\].

Step 4: Recognize the left-hand side as a derivative

The left-hand side of the equation \(e^{2x} \frac{dy}{dx} + 2e^{2x}y\) is the derivative of \(y e^{2x}\) with respect to \(x\), that is \(\frac{d}{dx}(ye^{2x})\). Rewrite the equation:\[\frac{d}{dx}(ye^{2x}) = xe^{2x}\].

Step 5: Integrate both sides

Integrate both sides of the equation with respect to \(x\):\[\int \frac{d}{dx}(ye^{2x}) \, dx = \int xe^{2x} \, dx\].The left side simply integrates to \(ye^{2x}\). The right side requires integration by parts.

Step 6: Integrate the right side using integration by parts

For integration by parts, let \(u = x\) and \(dv = e^{2x} dx\). Then \(du = dx\) and \(v = \frac{1}{2}e^{2x}\). The integral becomes \(\frac{1}{2}xe^{2x} - \frac{1}{2}\int e^{2x} dx = \frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x}\).

Step 7: Solve for \(y\)

Substitute back into the integrated equation: \[ye^{2x} = \frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x} + C\].Divide every term by \(e^{2x}\) to solve for \(y\):\[y = \frac{1}{2}x - \frac{1}{4} + Ce^{-2x}\], where \(C\) is the constant of integration.

Key Concepts

first-order linear differential equations

First-order linear differential equations are a prevalent type of differential equation often seen in calculus and engineering courses. These are characterized by the presence of a first derivative and no higher derivatives. They can be generally expressed in the form: \[ \frac{dy}{dx} + P(x)y = Q(x) \]where:

• \(P(x)\) and \(Q(x)\) are functions of \(x\).
• The solution involves both integration and algebraic manipulation.

This form is quite important because it shows a direct relationship between the variable \(y\) and its rate of change. Understanding this standard form enables us to apply specific methods to find solutions, such as using integrating factors.

integrating factors

An integrating factor is a clever mathematical tool that simplifies first-order linear differential equations. It transforms a difficult equation into one that can be integrated more easily. The integrating factor, \(\mu(x)\), is determined through the formula: \[ \mu(x) = e^{\int P(x) \, dx} \]Here's how it works:

• Multiply the entire differential equation by the integrating factor \(\mu(x)\).
• This results in the left-hand side becoming the derivative of a product, making it straightforward to integrate.
• In our example, the integrating factor is \(e^{2x}\), which simplifies the equation significantly.

Using integrating factors is a key technique in tackling these equations, turning complex problems into manageable ones.

integration by parts

Integration by parts is a powerful technique used to evaluate integrals that are products of functions. It is particularly useful when direct integration is difficult. The formula for integration by parts is derived from the product rule for differentiation and is given by:\[ \int u \, dv = uv - \int v \, du \]where:

• Choose \(u\) and \(dv\) from the integral you need to solve.
• Calculate \(du\) and \(v\), remembering that \(v\) is found by integrating \(dv\).
• Apply the formula and solve the resulting integrals.

In our exercise, this method was used to integrate the term \(xe^{2x}\), which otherwise would be challenging to solve directly.

constant of integration

In the context of differential equations, each indefinite integral introduces a constant of integration, usually denoted by \(C\). This constant is crucial since it represents the family of solutions to the differential equation.

• When integrating, we determine the general solution which includes \(C\).
• This constant accounts for all possible vertical shifts of the solution curve on a graph.
• In specific applications, an initial condition allows us to solve for \(C\), yielding a particular solution.

In our example, the constant of integration appears when we integrate both sides. Once the general form is found, you might need additional information to find the specific value of \(C\) that satisfies given initial conditions.