$\\mathrmLHS=y^\\prime \\prime+y=-\\frac12(-x \\cos x-2 \\sin x)+\\left(-\\frac12 x \\cos x\\right)=\\sin x=\\mathrmRHS,$ so $y=-\\frac12 x \\cos x$ is a solution of the differential equation.
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we desire to know which the the following functions are options City differentially freaking. For this reason this is ours front differential equation. So we desire to uncover our wide dual primed and also our wives and also put those in Syria. For this reason let's begin with a Tony too much and vast over prime. For this reason first, let's remember that the derivative Ah, cause, uh is negative, son. And also the derivative of authorize is confident co sign X. So simply some weird interviews. Do you remember? therefore let's take it artery the sign we're gonna get coastline Onyx and also I defended a room that we get an unfavorable sign. It's for this reason we're just gonna placed those right into our equation here so we'll have an unfavorable sign picks plus sign X world to sign X and an ext left and saw. This is zero, and also it's a authorize X These space not equal. Therefore, we know A is not our answer for bee. We're under the same thing to define what crime, i beg your pardon is negative. Signed X and then we're gonna obtain our wide twin product, which would be bring out our an unfavorable review and also sign is co sign so negative co sign it. Placed those right into our equation, will certainly have negative co signed X you re welcome Costa top top X, you carry down our best side. Far left side of zero will know Right next is the sine X and also these are not the same. Thus be Zlata. Prize Joyce Fighter Chrissy, we require to find what crimes in one dual prime location or equation. Therefore it's Yvonne. Why prime? We're going to use product will. So exactly how was the very first times that a evaluation of the second close? The circuit times the derivative with first we said prior to that will have one have actually that. Five sonic close one have actually signed it. We're gonna perform the very same thing to find my double prime. We have to use product dominion for this section here. So us say the very first times that ever before got a second plus the second times that during the very first it's simply one upside and also they to be students. Carry down this part here, but it's derivative and also not derivative is one have actually co sign it. So us simplify. This will have negative one happy ex tonics plus 1/2 co authorize IX. That's among those on exception one ex villages, yet no one here or interpreted one. There So currently we have actually We're why don't most likely equation and r y equation and you just plug those right into our function. We'll have negative 1/2 ex Sonics, close coastline onyx list or while equation and bring under our right hand side. So we simplify. We have a an unfavorable 1/2 X I following year on a optimistic here. So just counsel v each other and also we have actually co sign X equals sine X over. This is no true. Therefore, grab letter answer playthings either Does the look a dead day? So even though we understand that this has been an answer toys, we can still occupational it out. For this reason we have our wives functions. I need to uncover our wives crime. So it's best there. You speak the very first times the ribbon that the second plus the spec that times that you're among the an initial possible bodice will have actually positive 1/2 discover in it. Wilderness won she goes ~ above it. Never use this upon ours war dual from equation. So take it the first times that eliminated a second plus the 2nd times that throughout your v the very first and ns was the derivative. This portion. Now it's favor through this portion. So we have actually a minus one have sine X, but it changes. Sign since they removed can sign in the an adverse sign. However we simplify this. We'll have a one have actually ex co signed X and then just be one happens when fifty percent that's simply one. So but signing? So currently we have actually off the equation. So we need to plug into our differential equation here. Threw in a pregnancy. For this reason you have actually want, huh? Thanks. Co authorize it. Let's discover it. Add to a an adverse one, huh? No. Co authorize it and also bring down our best hand side. Definitely. Look here and also here. This subtract the end they were left with signed X is may be to sign it. Therefore, D is ours answer choice.