/*! 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 23 Determine the quadrants in which... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 23

Determine the quadrants in which the solution of the differential equation is an increasing function. Explain. (Do not solve the differential equation.) $$ \frac{d y}{d x}=\frac{1}{2} x y $$

Short Answer

Expert verified
The solutions of the differential equation are increasing in the 1st and 3rd quadrants.

Step by step solution

01

Identify The Variables

The differential equation given is \(\frac{dy}{dx}=\frac{1}{2} xy\). Here, \(\frac{dy}{dx}\) represents the rate of change of the function \(y\) with respect to \(x\), \(x\) is the independent variable and \(y\) is the dependent variable.
02

Determine the Sign of the Derivative

To find where the function \(y\) is increasing, we need to find the areas where the derivative (represented by the right-hand side of the equation, \(\frac{1}{2}xy\)) is positive. The product of \(x\) and \(y\) will be positive in the 1st quadrant (where both \(x\) and \(y\) are positive) and in the 3rd quadrant (where both \(x\) and \(y\) are negative).
03

Final Conclusion

From the previous step, we determined that the solution is increasing in the 1st and 3rd quadrants. Thus, these are the quadrants in which the solution to the given differential equation is an increasing function.

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!

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

The area of the region bounded by the graphs of \(y=x^{3}\) and \(y=x\) cannot be found by the single integral \(\int_{-1}^{1}\left(x^{3}-x\right) d x\). Explain why this is so. Use symmetry to write a single integral that does represent the area.

Fluid Force on a Rectangular Plate A rectangular plate of height \(h\) feet and base \(b\) feet is submerged vertically in a tank of fluid that weighs \(w\) pounds per cubic foot. The center is \(k\) feet below the surface of the fluid, where \(h \leq k / 2\). Show that the fluid force on the surface of the plate is \(\boldsymbol{F}=w k h b\)

A tank on a water tower is a sphere of radius 50 feet. Determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. (Note: Use the zero or root feature of a graphing utility after evaluating the definite integral.)

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places. $$ y=x^{2}, \quad y=4 \cos x $$

Use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\ln x, \quad y=0, \quad x=1, \quad x=3 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

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.