/*! 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 52 Finding a General Solution In Ex... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 52

Finding a General Solution In Exercises \(43-52\) , use integration to find a general solution of the differential equation. $$\frac{d y}{d x}=5(\sin x) e^{\cos x}$$

Short Answer

Expert verified
The general solution is \(y(x) = -5e^{\cos(x)} + C\).

Step by step solution

01

Identify Integral

We first identify that the integral of this function is required to find the solution to this differential equation. The given function is \(y'(x)=5(\sin x) e^{\cos x}\). This can be solved as an integral problem, \(\int y'(x) dx = \int 5(\sin x) e^{\cos x} dx\).
02

Integral Solution

Next is the integration process. This integral can be solved via substitution method. Let \(u = \cos(x)\), then \(du = -\sin(x) dx\). By rearranging, we'll get \(- du = \sin(x) dx\). Substituting into the integral gives: \(-5 \int e^{u} du\). Taking the integral results in \(-5e^u\).
03

Back Substitution

Now, revert the substitution to get the solution in terms of \(x\). Replace \(u\) by \(\cos(x)\). This gives: \(-5e^{\cos(x)}\) as the integral/solution.
04

General Solution

Finally, don't forget that when you integrate, you should add the constant of integration. So the complete solution is \(y(x) = -5e^{\cos(x)} + C\), 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.

Understanding Integration Techniques
When working with differential equations, integration becomes an essential tool. Integration essentially helps us to reverse differentiation, allowing us to find functions when their rate of change is given. In the context of the exercise, to solve the equation \( \frac{dy}{dx} = 5(\sin x) e^{\cos x} \), we integrate:
  • The derivative \( y'(x) \) can be integrated to find \( y(x) \).
  • The integral \( \int 5(\sin x) e^{\cos x} \, dx \) captures the behavior of the original function.
Often, integration requires specific techniques to simplify the process. Recognizing patterns and transformations can be key to efficiently solving complex integrals.
Mastering the Substitution Method
The substitution method is an integration technique often used to simplify integrals by making a clever choice of a substitution variable. In the given problem, the integral \( \int 5(\sin x) e^{\cos x} \, dx \) can be tricky. The substitution method works by:
  • Selecting a part of the integrand to substitute with a new variable \( u \). Here, \( u = \cos(x) \).
  • Finding the differential \( du \), which in this case is \( du = -\sin(x) \, dx \).
  • Rewriting the integral in terms of \( u \): \(-5 \int e^u \, du \).
This transformation simplifies the integral, allowing us to focus on the more straightforward function \( e^u \). After integrating, the substitution is reversed to express the solution back in terms of the original variable \( x \).
Importance of the Constant of Integration
Whenever you perform indefinite integration, adding a constant of integration \( C \) is vital because it represents the family of solutions. This constant is necessary to account for all the possible functions that could have the same derivative:
  • After integrating \(-5 \int e^u \, du\), we reach \(-5e^u + C\).
  • Substituting back for \( u = \cos(x) \), we get the general solution: \( y(x) = -5e^{\cos(x)} + C \).
This constant ensures that all variations of the function that have the same slope are accounted for, capturing the complete set of solutions to the differential equation.

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

Using a Logistic Differential Equation In Exercises 55 and 56, the logistic differential equation models the growth rate of a population. Use the equation to (a) find the value of \(k,(\) b) find the carrying capacity, (c) graph a slope field using a computer algebra system, and (d) determine the value of \(P\) at which the population growth rate is the greatest. $$\frac{d P}{d t}=0.1 P-0.0004 P^{2}$$

Carbon Dating Carbon-14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of \(^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \(^{14} \mathrm{C}\) absorbed by a tree growing today. A piece of ancient charcoal contains only 15\(\%\) as much of the radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal? (The half-life of \(^{14} \mathrm{C}\) is 5715 years.)

Population In Exercises \(51-54,\) the population (in millions) of a country in 2015 and the expected continuous annual rate of change \(k\) of the population are given. (Source: U.S. Census Bureau, International Data Base) (a) Find the exponential growth model \(P=C e^{k t}\) for the population by letting \(t=5\) correspond to \(2015 .\) (b) Use the model to predict the population of the country in \(2030 .\) (c) Discuss the relationship between the sign of \(k\) and the change in population for the country. \begin{array}{l}{\text { Country }}&{\text { 2015 Population }}&&{\text { \(k\) }} \\ {Paraguay}& {\text { 6.8}}&&{{0.012}}\end{array}

Determining if a Function Is Homogeneous In Exercises \(69-76,\) determine whether the function is homogeneous, and if it is, determine its degree. A function \(f(x, y)\) is homogeneous of degree \(n\) if \(f(x, t y)=t^{n} f(x, y) .\) $$f(x, y)=x^{4}+2 x^{2} y^{2}+x+y$$

A not uncommon calculus mistake is to believe that the product rule for derivatives says that \((f g)^{\prime}=f^{\prime} g^{\prime} .\) If $$f(x)=e^{x^{2}}$$ determine, with proof, whether there exists an open interval \((a, b)\) and a nonzero function \(g\) defined on \((a, b)\) such that this wrong product rule is true for \(x\) in \((a, b)\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.