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Numerade Educator

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Problem 5 Medium Difficulty

Which of the following functions are solutions of the differential equation $ y^{"} + y = \sin x ? $

(a) $ y = \sin x $
(b) $ y = \cos x $
(c) $ y = \frac {1}{2} x \sin x $
(d) $ y = - \frac{1}{2} x \cos x $

Answer

$\mathrm{LHS}=y^{\prime \prime}+y=-\frac{1}{2}(-x \cos x-2 \sin x)+\left(-\frac{1}{2} x \cos x\right)=\sin x=\mathrm{RHS},$ so $y=-\frac{1}{2} x \cos x$ is a solution of the differential equation.

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Video Transcript

we want to know which of the following functions are solutions City differentially freaking. So this is our front differential equation. So we want to find our wide double primed and our wives and put those in Syria. So let's start with a Tony too far and wide over prime. So first, let's remember that the derivative Ah, cause, uh is negative, son. And the derivative of Sign is positive co sign X. So just some odd interviews. Do you remember? So let's take artery the sign we're gonna get Coast Onyx and I protected a room that we get negative sign. It's so we're just gonna put those into our equation here so we'll have negative sign picks plus sign X people to sign X and more left and saw. This is zero, and it's a sign X These are not equal. Therefore, we know A is not our answer for bee. We're under the same thing to define what crime, which is negative. Signed X and then we're gonna get our wide double product, which would be bring out our negative review and sign is co sign so negative co sign it. Put those into our equation, will have negative co signed X Please Costa on X, you bring down our right side. Far left side of zero will know Right side is the sine X and these are not the same. Therefore be Zlata. Answer Joyce Fighter Chrissy, we need to find what crimes in one double prime placement or equation. So it's Yvonne. Why prime? We're going to use product will. So how was the first times that a review of the second close? The circuit times the derivative with first we said before that will have one have that. Oh sonic close one have signed it. We're gonna do the same thing to find my double prime. We have to use product rule for this portion here. So we say the first times that ever got a second plus the second times that during the first it's just one upside and they were students. Bring down this portion here, but it's derivative and not derivative is one have co sign it. So we simplify. This will have negative one happy ex tonics plus 1/2 co sign IX. That's one of those on exceptions one ex villages, but no one here or understood one. There So now we have We're why don't probably equation and r y equation and you just plug those into our function. We'll have negative 1/2 ex Sonics, close coast onyx lists or while equation and bring down our right hand side. So we simplify. We have a negative 1/2 X I next year on a positive here. So just counsel with each other and we have co sign X equals sine X over. This is not true. Therefore, seize letter answer toys either Does it look a Shame day? So even though we know that this has been an answer toys, we could still work it out. So we have our wives functions. I need to find our wives crime. So it's right there. You say the first times the ribbon of the second plus the spec It times that you're one of the first possible bodice will have positive 1/2 explore in it. Wilderness won her goes on it. Never use this upon our war double from equation. So take the first times that removed a second plus the second times that during your with the first and I was the derivative. This portion. Now it's like through this portion. So we have a minus one have sine X, but it changes. Sign because they removed could sign in the negative sign. But we simplify this. We'll have a one have ex co signed X and then just be one happens when half that's just one. So but signing? So now we have off the equation. So we need to plug into our differential equation here. Threw in a pregnancy. So you have want, huh? Thanks. Co sign it. Let's find it. Plus a negative one, huh? No. Co sign it and bring down our right hand side. Definitely. Look here and here. These subtract out they were left with signed X is able to sign it. Therefore, D is our answer choice.