Chapter 8: Problem 7

Find the general solution of each of the following differential equations. \(\left(1+e^{x}\right) y^{\prime}+2 e^{x} y=\left(1+e^{x}\right) e^{x}\)

Short Answer

Expert verified

The general solution is \( y = \frac{(1+e^x)}{3} + \frac{C}{(1+e^x)^2} \).

Step by step solution

Identify the type of differential equation

The given differential equation is \(\left(1+e^{x}\right) y^{\prime}+2 e^{x} y=\left(1+ e^{x}\right) e^{x}\). This is a first-order linear differential equation in the form: \( a(x)y' + b(x)y = c(x) \), where \( a(x) = 1 + e^x \), \( b(x) = 2e^x \), and \( c(x) = (1 + e^x) e^x \).

Rewrite the differential equation in standard form

Standard form for a first-order linear differential equation is: \( y' + P(x)y = Q(x) \). Divide both sides by \(1 + e^x \):\[ y' + \frac{2e^x}{1+e^x} y = \frac{(1+e^x) e^x}{1+e^x} \]. Simplify to get: \[ y' + \frac{2e^x}{1+e^x} y = e^x \].

Identify integrating factor

The integrating factor \( \mu(x) \) is determined by \( \mu(x) = e^{\int P(x) dx} \). Here, \( P(x) = \frac{2e^x}{1+e^x} \). Find the integral of \(P(x) \): \[ \int \frac{2e^x}{1+e^x} dx \]. Let \( u = 1+e^x \) then \( du = e^x dx \). The integral becomes \[ \int \frac{2du}{u} = 2 \ln|u| = 2 \ln|1+e^x| \]. Thus, \( \mu(x) = e^{2 \ln|1+e^x|} = (1+e^x)^2 \).

Multiply through by the integrating factor

Multiply the entire differential equation by the integrating factor \( (1+e^x)^2 \) to get: \[ (1+e^x)^2 y' + \frac{2e^x (1+e^x)}{1+e^x} y = (1+e^x)^2 e^x \]. This simplifies to: \[ (1+e^x)^2 y' + 2e^x (1+e^x) y = (1+e^x)^2 e^x \].

Recognize and simplify left-hand side

Notice that the left-hand side of the equation is a derivative of the product of the integrating factor and y: \[ \frac{d}{dx}\left[ (1+e^x)^2 y \right] = (1+e^x)^2 e^x \].

Integrate both sides

Integrate both sides with respect to x: \[ \int \frac{d}{dx}\left[ (1+e^x)^2 y \right] dx = \int (1+e^x)^2 e^x dx \]. The left-hand side becomes \[ (1+e^x)^2 y \]. For the right-hand side, let \( u = 1+e^x \), then \( du = e^x dx \): \[ \int u^2 du = \frac{u^3}{3} = \frac{(1+e^x)^3}{3} + C \].

Solve for y

Divide both sides by \( (1+e^x)^2 \) to solve for \( y \): \[ y = \frac{(1+e^x)^3/3 + C}{(1+e^x)^2} = \frac{(1+e^x)}{3} + \frac{C}{(1+e^x)^2} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integrating Factor

An integrating factor is a function used to multiply through a differential equation to make it easier to solve. This technique is particularly useful for first-order linear differential equations.

For a differential equation of the form:

\( y' + P(x)y = Q(x) \), the integrating factor \( \mu(x) \) is given by:

\( \mu(x) = e^{\int P(x) dx} \).

This integrating factor, when multiplied with the entire equation, helps to transform the left-hand side into the derivative of a product of the integrating factor and \( y \).

In our example, \( P(x) = \frac{2e^x}{1 + e^x} \). Integrating \( P(x) \), we get:

\( \int \frac{2e^x}{1 + e^x} dx = 2 \ln|1 + e^x| \. \)

Thus, the integrating factor:

\( \mu(x) = e^{2 ln|1 + e^x|} = (1 + e^x)^2 \).

Differential Equations

Differential equations involve derivatives and are used to model various real-world phenomena like population growth, heat conduction, and motion. The equation in our exercise is a first-order linear differential equation of the form:

\( (1 + e^x)y' + 2e^x y = (1 + e^x)e^x \. \)

Here:

\( y' \) represents the first derivative of \( y \)
\( \left(1+e^x\right) \) and \( 2e^x \) are functions of \( x \) that multiply \( y' \) and \( y \) respectively
\( (1 + e^x) e^x \) is the function on the right side of the equation

The goal is to find a function \( y(x) \) that satisfies this equation. By following systematic solution techniques, we can uncover this function.

Solution Techniques

To find the solution of a first-order linear differential equation, we employ specific techniques. The main steps are:

Rewrite the equation in standard linear form \( y' + P(x)y = Q(x) \)
Identify and calculate the integrating factor \( \mu(x) \)
Multiply the entire equation by the integrating factor
Recognize and simplify the resulting expression
Integrate both sides to find the solution

In our specific example, after calculating the integrating factor:

\( (1 + e^x)^2 \, \) we multiply it throughout the differential equation, transforming it into:

\( \frac{d}{dx}[(1 + e^x)^2 y] = (1 + e^x)^2 e^x \. \)

Upon integration and subsequent simplification, we ultimately derive the general solution:

\( y = \frac{1 + e^x}{3} + \frac{C}{(1 + e^x)^2} \. \)

This systematic approach helps to find the solution efficiently and correctly.