/*! 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 48 Find a differential equation tha... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 48

Find a differential equation that is not separable.

Short Answer

Expert verified
Consider the differential equation \(y'(x) + y(x)\cos(x) = \sin(x)\). Upon inspection, we see that it is not possible to separate the dependent variable \(y(x)\) and the independent variable \(x\), nor can we rewrite it in the form \(M(x,y) dx + N(x,y) dy = 0\). Therefore, this differential equation is not separable.

Step by step solution

01

Consider the following differential equation

\(y'(x) + y(x)\cos(x) = \sin(x)\)
02

Check for Separability

In order to determine if this differential equation is separable, we need to check if it can be written in the form \(\frac{dy}{dx} = G(x)H(y)\) or \(M(x,y) dx + N(x,y) dy = 0\), where \(G(x)\) and \(H(y)\) are functions of only \(x\) and \(y\), respectively, and the same for \(M(x,y)\) and \(N(x,y)\). Rewrite the given differential equation as: \[\frac{dy}{dx} = -y(x) \cos(x) + \sin(x)\] We can see that the right-hand side of the equation has terms including both \(y(x)\) and \(x\) so we cannot write the differential equation in the form \(\frac{dy}{dx} = G(x)H(y)\). Moreover, we cannot separate the variables in the form of \(M(x,y) dx + N(x,y) dy = 0\) since we cannot express each term with one variable only. Hence, the original differential equation is not separable.

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.

Separable Differential Equations
Separable differential equations are a special class of differential equations that allow us to split up variables and solve them more easily. They typically look like \[ \frac{dy}{dx} = G(x)H(y) \]where we can separate the function of \(x\) from the function of \(y\). This makes integration of both sides possible.When dealing with separable differential equations, the approach often involves rearranging terms to isolate \(y\) and \(dy\) on one side, and \(x\) and \(dx\) on the other. Here鈥檚 a quick guideline:
  • Identify if the differential equation can be expressed in a separable form.
  • Rearrange the equation to move \(y\) and \(dy\) to one side and \(x\) and \(dx\) to the other.
  • Integrate both sides separately.
  • Don鈥檛 forget to include the constant of integration!
This process can provide a straightforward strategy to finding solutions, but this luxury is not always available with every differential equation.
Non-separable Equations
Non-separable differential equations require a bit more ingenuity, as they cannot be split into parts that only involve single variables. Our example,\[y'(x) + y(x)\cos(x) = \sin(x)\]demonstrates this, as no rearrangement can isolate \(x\) or \(y\) for separation. For non-separable equations, alternative methods must be applied, such as integrating factors or transforming the equation.

Why Can't This Equation Be Separated?

Sometimes, the presence of both \( y(x) \) and terms that include \( x \) means there is an art to finding solutions. Methods often involve:
  • Examining substitutions: Attempt substitutions that might simplify the equation.
  • Using specific techniques: Integrating factors can sometimes transform an equation into a manageable form.
Understanding these strategies expands your toolset for tackling complex problems that go beyond simple separability.
Calculus Techniques
Calculus offers a variety of techniques to solve differential equations, especially those that aren鈥檛 neatly separable.

Using Integrating Factors

An integrating factor is often used for linear first-order equations to facilitate integration. This technique involves:
  • Multiplying through by a carefully chosen function to make the equation integrable.
  • Transforming the equation to use derivatives of products, simplifying the integration process.
This technique is particularly useful for linear equations that are not separable.

Exploring Substitution

Sometimes applying clever substitution can make an otherwise complex equation simpler:
  • Try substituting \( y \) or \( x \) with another variable to simplify the terms.
  • Identify patterns that resemble known integrable forms, and manipulate them to fit.
These calculus techniques provide versatile tools, helping to navigate the complexities of various differential equation types. They open pathways toward solutions that might initially seem elusive.

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

Decide whether or not the given integral converges. If the integral converges, compute its value. $$\int_{-\infty}^{-2} \frac{1}{x^{2}} d x$$

For each differential equation, find the particular solution indicated. HINT [See Example 2b.] $$\frac{d y}{d x}=\frac{x y}{\left(x^{2}+1\right)^{2}} ; y(0)=1$$

Chocolate Mousse Sales The weekly demand for your company's Lo-Cal Mousse is modeled by the equation \(q(t)=\frac{50 e^{2 t-1}}{1+e^{2 r-1}}\) gallons per week where \(t\) is time from now in weeks. Investigate the integrals \(\int_{0}^{+\infty} q(t) d t\) and \(\int_{-\infty}^{0} q(t) d t\) and interpret your answers.

Decide whether or not the given integral converges. If the integral converges, compute its value. $$\int_{-\infty}^{2} e^{x} d x$$

It sometimes happens that the Fundamental Theorem of Calculus gives the correct answer for an improper integral. Does the FTC give the correct answer for improper integrals of the form $$ \begin{aligned} &\quad \int_{-a}^{a} \frac{1}{x^{1 / r}} d x \\ &\text { if } r=3,5,7, \ldots ? \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.