/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Finding a General Solution In Ex... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 44

Finding a General Solution In Exercises \(41-52,\) use integration to find a general solution of the differential equation. $$ \frac{d y}{d x}=\frac{e^{x}}{4+e^{x}} $$

Short Answer

Expert verified
The general solution of the given differential equation is \(4y = x - 4 \ln |4+e^{x}| + C\).

Step by step solution

01

Separate variables

Firstly, aim to separate the variables so that \(y\) and \(x\) are on different sides of the equation. Transform the equation as: \(4 \cdot dy =(e^{x} \cdot dx) / (4+e^{x})\)
02

Additional transformations

Next, perform some additional transformations, writing \(e^{x}\) as \((4+e^{x}) - 4\), which makes the right side of equation become \(4 \cdot dy = ((4+e^{x})-4 \cdot dx)/ (4+e^{x})\), or \(4 \cdot dy = dx - 4 \cdot dx / (4+e^{x})\)
03

Integrate both sides

Now integrate both sides of the equation. This results in \(4y = x - 4\int dx/(4+e^{x})\). The integral on the right side could be solved with the substitution method, by setting \(u =4+e^{x}\), so \(du = e^{x}dx\)
04

Complete the solution

To find the general solution, calculate the integral and return to the variable \(x\). The complete solution is \(4y = x - 4 \ln |4+e^{x}| + C\), where \(C\) is the constant of integration.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Integration
Integration is a fundamental concept in calculus often used to find solutions to differential equations. The idea is to find the antiderivative of a function, which generally means reversing the process of differentiation. Understanding integration helps us determine the solution to a differential equation by finding a function whose derivative is the given function.

For this particular exercise, integration is applied after transforming the original differential equation into a form where the variables can be separated. The result of the integration will include an arbitrary constant, often denoted as \( C \), because integration represents an indefinite process that yields multiple potential solutions.
  • Integral calculus reverses differentiation.
  • General solutions include a constant \( C \).
Separation of Variables
Separation of Variables is a technique used in solving differential equations, where we separate the variables \( y \) and \( x \), pushing them on opposite sides of the equation. This allows each side of the equation to be linked solely with one variable, providing a pathway to solve the equation using integration.

In the given exercise, initially, the equation \( \frac{d y}{d x} = \frac{e^{x}}{4+e^{x}} \) is manipulated so that all terms involving \( y \) are on one side, and all terms involving \( x \) are on the other.
  • Separate \( dy \) and \( dx \) onto different sides.
  • Prepare the equation for integration.
Substitution Method
The substitution method is a powerful tool in integrating complex functions. This method involves substituting a part of the integral with a new variable \( u \), simplifying the equation and making it more accessible.

In our exercise, after transforming the equation into a separable form, substitution is used to integrate the expression \( \int \frac{dx}{4+e^{x}} \). To do this, we set \( u = 4 + e^{x} \) and find \( du = e^{x} dx \). This substitution simplifies the integral into a more straightforward form, \( \int \frac{1}{u} du \), which then can be easily integrated to \( \ln |u| \).
  • Replace an expression with a single different variable \( u \).
  • Simplify the integral for easier computation.
  • Don't forget to switch back to the original variable after integration.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determining if a Function Is Homogeneous In Exercises \(67-74,\) determine whether the function is homogeneous, and if it is, determine its degree. A function \(f(x, y)\) is homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y)\) $$ f(x, y)=\tan \frac{y}{x} $$

Separation of Variables In your own words, describe how to recognize and solve differential equations that can be solved by separation of variables.

Finding a General Solution Using Separation of Variables In Exercises \(1-14,\) find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{6-x^{2}}{2 y^{3}} $$

Solving a First-Order Linear Differential Equation In Exercises \(51-58\) , solve the first-order differential equation by any appropriate method. $$ y \cos x-\cos x+\frac{d y}{d x}=0 $$

Finding a Particular Solution Using Separation of Variables In Exercises \(15-24\) , find the particular solution that satisfies the initial condition. $$ d T+k(T-70) d t=0 \quad T(0)=140 $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.