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What are the integrating factors for the following differential equations?$$y^{\prime}+\ln x=\frac{y}{x}$$

$y=\frac{x \ln ^{2}(x)}{2}+C x$

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

Harvey Mudd College

Idaho State University

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

10:23

What are the integrating f…

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it's real. We are going to solve this differential equation why prime minus Y over X. Equal minus lock eggs. So this is standard form. We first find integrating factor The P. Rx is -1 over X. So we integrate -1 over XDX. This is going to become minus a lot of absolute X. Right? Which cancer with E. L. Become one over X absolute eggs. So we multiply this true on both sides. And since we have absolute value on both sides we can treat it as normal polynomial. Because in the case that is is It appeared as -1 It's essentially just having -1 multiply both side of the equation and it cancel each other out. So we treated as just polynomial. The left hands are going to become white times one over X. Of this prime. And now we just need to integrate the right hand side. And luckily this time even if we have block X here we can easily substitute. Why? As lock eggs. And sorry substitute us lock eggs. And do you gonna Far on this form one over X. DX which is exactly what we have. So by this substitution we have in the goal you D. You so we get new square Over two just like that now. Okay we have minus here. So is minus you square or what to plus a constant. And we just put this back and times X. To the right side to get the answer. Who's gonna be X. Times oh minus lock X. Square over to plus yeah C times X. Right. Mhm. And that is it? This is the answer. Thank you.

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