Problem

What are the integrating factors for the followin…

04:17

Problem 226 Hard Difficulty

What are the integrating factors for the following differential equations?
$$
(x+2) y^{\prime}=3 x+y
$$

Answer

$y=3(x+2) \ln |x+2|+6+(x+2) C$

Related Courses

Calculus 2 / BC

Calculus

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

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Video Transcript

we are going to solve this differential equation. Why prom minus Y over X plus two equal three X over X plus two. This is the standard form. So first step we gonna find integrating factors Here P of X is -1 over X plus two. Right? So we integrate this the eggs and power E. To this amount we're going to get minus Long, absolute x plus two over eat. And this is lock is on base E. Right? So we can cancel it out This become one over Absolute X-plus two now. Mhm. The absolute value may looks complicated at first but in this case it can be easily there with because we are multiplying both sides of the equation with this amount. So yeah, you're gonna see in a minute why is not a big deal when we first multiply this true, we're going to have absolute value left over like this. But then when you consider all the cases possible when when absolutely a little give plus on minus side you're going to see that on a case where they are minus is essentially our original equation multiply -1 on both sides. Right? That means this -1, cancel each other out in the end. So no matter what value of X is we can always make it so that we get the positive side because minus one can be canceled on both sides. This is going to become just pulling a meal X plus two square. And so we can easily stop this. So left side gonna collapsed too. Yes, prime. So why times X plus two, prime equals three X over X plus two square. And now if we can integrate the right hand side, we're gonna help. Why? So let's do it. This is also not complicated. We use substitution let you be X plus two then the you is just the X. Right? So we share this too in the grade three export B tree of you minus two. Right? Or were you square? Yeah and Kia D. You So this is polynomial. We can Break it into two parts tree over you minus uh six over your square. All right. And both of them just holding the meal so we get three love absolute you minus This will be plus six over you. Right? Because integrated when we use choir We get -1. So this is the right hand side And you is just X-plus two here. So now we can go back and see that all we have to do is multiply X plus two two. That amount that we just get and the answer will be this value. Mhm Don't forget the constant term here. I think I forget to write it down. Okay do not forget the constant term here. And that is it. This is absolutely X plus two. And this is the answer. Thank you

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Gilbert Strang,

Calculus

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