Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

This exercise generalizes Exercise $31 .$ Suppose, $\frac{d y}{d x}=f\left(\frac{y}{x}\right) ;$ show that the change of variable $v=y / x$ results in a separable differentiable equation.

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Campbell University

Harvey Mudd College

Baylor University

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

03:06

Show that the change of va…

01:43

Suppose that $y$ is a diff…

00:32

00:41

02:53

Assertion (A): If two diff…

00:36

Suppose $F(x)=x^{3} f(x),$…

00:40

with this problem, we've been asked to show for a general differential equation of the form in the top left under the variable transformation of evils X y. This becomes a separable differential equation in the variables v. X. So we'll start by looking at the left hand side and right hand sides separately. So the left hand side, the first thing we want to do is rewrite or variable transformation in terms of why we do this by dividing both sides by X, receive that y equals the Kovacs. Now also, just write this in index imitation. So we have indices involved because I find it's easier when we're using the product rule. So we want to find d Y. The X on will use the product rule on this y equals V times. Excellent minus one. And this gives us. But due by the X is equal to X the minus one DVD X minus B X over minus two. And if we just put this back into fractional notation, this is one over X DVD X minus V over X squared. This is the left hand side of our differential equation. So, you know, look at the right hand side we have Why? Over x times A Function of X, My now ex wife, Simply V saver function becomes a function of the and we've already seen why is equal to D of Rex. So why over act becomes the over X squared and that's the right hand side of our equation. Person. This together or differential equation now reads. Want over X DVD X minus the over X squared that's equal to be over X squared times function of the We can simplify this first by multiplying each time by X gives us Devi over the X minus. The over X equals the ever X function of the and take the turn the over X to the right hand side. We get that DVD X is equal to the over X times one plus function fee. Now the final form we want to get this in is to just bring the V variables of the left hand side on receiving fister Friendship equation inseparable and is equal to one over V times one plus functionality DVD X, and that is equal to one over X. This is the final form we are asked to share

View More Answers From This Book

Find Another Textbook

Numerade Educator

01:34

Evaluate the given integral.$$\int 2 x\left(5^{4 x^{2}}\right) d x$$

03:13

The partial differential equation $c^{2} u_{x x}-u_{t t}=0$ where $c$ is a c…

01:14

A function is said to be homogeneous of degree $n$ if $f(\gamma x, \gamma y)…

02:34

We have that $\int \frac{1}{x} d x=\ln x+c,$ but it also follows from $\frac…

01:17

Determine the region $R$ determined by the given double integral.$$\int_…

02:16

Consider the area bounded by $f(x)=x^{2}+1$ and the $x$ -axis, between $x=0$…

01:52

Determine the area of the indicated region.Region bounded by $f(x)=2 / x…

01:33

Evaluate the given integral and check your answer.$$\int 2 t^{7} d t$$

04:12

Find (a) $f_{x x}(x, y),$ (b) $f_{y y}(x, y),$ (c) $f_{x y}(x, y),$ and $f_{…

(a) Compute $\frac{d}{d x}(x \ln x-x)$. (b) What function can you now antidi…