Problem

Solve the given differential equation. $$y^{\pri…

03:58

Problem 3 Medium Difficulty

Solve the given differential equation.
$$x^{2} y^{\prime}-4 x y=x^{7} \sin x, \quad x > 0$$

Answer

$x^{4}(\sin x-x \cos x+c)$

Related Courses

Calculus 2 / BC

Differential Equations and Linear Algebra

Chapter 1

First-Order Differential Equations

Section 6

First-Order Linear Differential Equations

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Video Transcript

Hello. Let's get to solving this problem here or goes to solve this manure to French first order differential equation. So step one is that we need to get into the following form. Why prime plus p of X of why is equal to queue of X. Okay, so step one, we need to divide everything by X squared. That way I will get rid of this X squared right here. And this will get us coasted form there you want. So we end up with the UAE prime because expletive of X rays one minus four When we do when we have the X over X squared, we end up with just the X on the bottom. So we ended before her ex. Why? And then finally, Exodus seven over X squared just leaves us with exit a five sine x. Okay, so now we just need to find the integrating factors because to solve any first order linear differential equation, we need to find an integrated factor in this case. Integrated factor will always be equal to e. It's in a row of p of X. So what is P of X? In this case, it's everything in front of the Why. So what's it from there? Why here is negative pore over X. So we just end up taking the integral of E to the negative four off the X hell To the integral of that. So integral of one over X is equal to Ellen X. We end up with E to the minus for L and X. Then we're going to use a rule of law Grooms That's pulls the minus four to the top. So we end up with E to the Ellen off X to the minus four appear there ago. Finally, we're taking the E to the logger base e, which is Ln so that it means everything is right here, just gets pulled down. So our answer for our integrated factor is just excellent minus four. But now I have to solve integral. We just need the most quite both sides by accident minus four, we end up with exit E minus four right here and of here Reina. But why Prime minus four over X. Why is equal to minus four times exit of 55 minutes? Forgiveness one we know but exited a sine x Okay, multiply both sides back to the minus four. So now we needed to the integral on both sides. So on the left hand side, we will always end up with wire times integrating factor. Because in this case, we're ending up with a product rule off white times of entering factor. So if you follow backwards, if you took the director of this right here, you end up worse off about their So the Cinderella dirt it gives back yourself. So meanwhile, on the right hand side, we need to do an integration by parts to solve this problem. So too did in English. My parts we just call the first part you call the second part D V. The formula is you v minus Integral sign Uh, the u times V. So since you, it's a no German plug and chug to find a parts you is X V is we don't know yet We need to take the integral of sine X, which is devi to find eat to find RV So the integral of sine X is equal to negative Croce Annex 13 X times negative coastline X minus. Do you do you is degenerative x A directive extra just one and we put back her integral sign. It's Times V, which is the integral sign X, which we already found its coastline X minus minus three is positive. So this ends up being a plus sign right here and we end up with Costa Annex. Okay, Very straightforward. So now we just need to the integral of Coastline X, which just gives us a sign, acts nothing special, and we end up with a concert in the end. So on the right hand side, we end up with X negative X Times Co sign X plus sine X plus C and finally only to do for my left hand side. We just need to multiply both sides by exit. Afford to get rid of that accident minus four. So we just end up of why is equal to x five. We have reminded stand right here co sign X plus exit of four because of motel and both sides by exit of four sine X plus C times exit before

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