/*! 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 1116 Find a first-order differential ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 43: Problem 1116

Find a first-order differential equation satisfied by all circles with center at the origin.

Short Answer

Expert verified
The first-order differential equation satisfied by all circles with center at the origin is: \( \frac{dy}{dx} = -\frac{x}{y} \)

Step by step solution

01

General equation of a circle centered at the origin

The general equation of a circle with radius R and centered at the origin (0,0) is given by: \(x^2 + y^2 = R^2\)
02

Differentiating with respect to x

Now, we will implicitly differentiate the equation with respect to x, since we are looking for a differential equation that involves the derivative: \(\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(R^2)\) Applying the power rule to differentiate, we get: \(2x + 2y \frac{dy}{dx} = 0\)
03

Simplify the equation

We can simplify the equation by dividing both sides by 2: \(x + y \frac{dy}{dx} = 0\)
04

Rearrange the equation

We can rearrange the equation to make \(\frac{dy}{dx}\) the subject, getting the final equation as: \(\frac{dy}{dx} = -\frac{x}{y}\) Thus, the first-order differential equation satisfied by all circles with center at the origin is: \(\frac{dy}{dx} = -\frac{x}{y}\)

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

A projectile is fired straight upwards with an initial velocity of \(1600 \mathrm{ft} / \mathrm{sec}\). What is its velocity at \(40,000 \mathrm{ft} .\) ? Assume \(g=32 \mathrm{ft} / / \mathrm{sec} .^{2}\)

When light radiation enters a medium, its rate of absorption with respect to the depth of penetration \(t\) is proportional to the amount of light that is incident on a unit area at that depth. Find the law relating \(\mathrm{L}\), the quantity of light, and \(\mathrm{t}\), if the incident light is \(\mathrm{L}_{0}\) and the emerging light after passing through a thickness \(\mathrm{t}_{1}\) is \(\mathrm{L}_{1}\).

A particle slides freely in a tube which rotates in a vertical plane about its midpoint with constant angular velocity w. If \(\mathrm{x}\) is the distance of the particle from the midpoint of the tube at time \(t\), and if the tube is horizontal with \(t=0\), show that the motion of the particle along the tube is given by \(\left(\mathrm{d}^{2} \mathrm{x} / \mathrm{dt}^{2}\right)-\mathrm{w}^{2} \mathrm{x}=-\mathrm{g} \sin \mathrm{wt}\). Solve this equation if \(\mathrm{x}=\mathrm{x}_{0}, \mathrm{dx} / \mathrm{dt}=\mathrm{v}_{0}\) when \(t=0 .\) For what values of \(\mathrm{x}_{0}\) and \(\mathrm{v}_{0} \mathrm{is}\) the motion simple harmonic?

A point moves on a curve in the \(\mathrm{x}-\mathrm{y}\) plane in such a way that the angle made by the tangent to the curve with the \(\mathrm{x}\) - axis is three times the angle between the radius vector and the \(\mathrm{x}\) -axis. Find the Cartesian equation of the family of curves satisfying this condition.

If the marginal cost of producing a certain item is \((d y / d x)=y^{\prime}=3+x+\left(e^{-\pi} / 4\right)\), what is the cost of producing 1 item if there is a fixed cost of \(\$ 4 ?\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Discrete 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.