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Question

Determine whether the given equation is the general solution or a particular solution of the given differential equation.
$$\frac{d y}{d x}+2 x y=0, \quad y=e^{-x^{2}}$$

Vikash Ranjan · Accepted Answer

In the problem we have what? S plus two X. Y equals zero. So do I. Upon The X is equal to -2 x. Y. Now immigration Divide upon white equal integration -2 x. The X. So it is Aaron Y. That is equal to -2 and works is called upon to. Let&#x27;s see. So why is equal to youtube or minus X Squire plus C. Or it is ce par minus exist clad. This is the gender solution. No, Here it is. Given Why equal you 4 -1 sq having no constant. So given it solution is of particular solution. This is the answer.

Smita Praharaj · Answer

this equation. In this question, we want to find a particular solution to the differential equation by prime equals to two x y, which passes through the 20.0 comma 1/2. And given that the general solution is why equals to see each of the X Square toe obtain the particular solution, all we need to split the point into the general solution. So if I so basically we need to substitute zero forex and 1/2 for why so we get 1/2 equals c times e to the zero square. Each of the zero is one, so we basically just get 1/2 equals to seek. So the particular solution y p equals 1/2 e to the X squared.

Mike Gaerlan · Answer

Okay, so we need to find solutions to this differential equation or a particular solution of this form. So not waiting to find why p prime of X. This is going to be the to a not e to the two X then why P double prime of X is equal to four a. Not each of the two X and then lastly, Y p triple prime of X is going to be able to eight a not e to the two X, then substituting these into this. Here we get eight a, not e to the two x plus and then also ate a not e to the two x minus to a not you, not e to the two x minus two a. Not be to the two x This is equal to four e to the two X combining all four of these terms. So this is 16 minus four. So we get 12 a, not e to the two X is equal to four. Eat the two x So this coefficient here 12 a not must be coefficient coefficient of it. So 12 a is equal to four or a not is going to be equal to 1/3. So four divided by 12 is 1/3. Okay, so our particular solution why P is going to be 1/3 e to the two X, like so, my p y p put of X lips. Why P of X is equal to that. It&#x27;s 1 32 the two X Next, we need to find a general solution, which means we also need to find the homogeneous solution to, uh which is equal to the homogeneous equation is this equation here is equal to zero. So we&#x27;re going to use, um, solutions of the form E to the, um are X. So that means then why Prime is equal to R E to the R X. Why double prime is equal to r squared E to the R X and then why a triple prime is equal to our cube. Eat the Rx. So now, plugging these into our equation Here we have our cube E to the r X plus two R squared E to the Rx minus. Um, are you to the Rx minus two? Ah, to be to the Rx is equal to zero next week. In fact, out in either the R X from all of these terms who get either the Rx and then our Q plus two r squared minus R minus two is equal to zero. So then now, since either the Rx cannot be equal to zero, then that means we just need to set this evil to zero. So to find the roots of that equation, I&#x27;m going to use synthetic division here. Okay, so according to, uh, the rational root beer. Um, since, um, this condition is one, the only factors when you consider are the roots of two. So in the plus or minus of those so plus or minus one plus or minus two. So I&#x27;m gonna start with one. So I bring down this one here. Okay, so then now one times one is one That gives us 31 times three is three ah, 13 minus one is going to be, too. And then two times one is two. That gets a zero. So that means one Here is a factor of this polynomial here so we can factor into our minus one and then r squared plus three are plus two nous to here is equal to zero this can further factor into, uh, so bring that down and then are plus one and R plus two is equal to zero, since the factors of two are just one and two and they both add up to three. So it factors into our plus one R plus two now, so the roots are going to be equal to one negative one and negative two, therefore are homogeneous. Solution is going to be equal to so why sea of X is equal to see one uh c one e to the x plus c two e to the negative x plus c three e to the negative two x So our general solution is going to be our homogeneous solution, plus our particular solutions. That&#x27;s illegal too. See one use of the X plus c two e to the negative x plus c three e to the negative two x and then plus 1/3 e to the two x

Smita Praharaj · Answer

So this problem we want to find particular solution of the differential equation. Why prime times X squared equals to buy. That passes through the 0.1 calmer to overeat. Given that the general solution is why equals to see times e to the negative one over X channel solution just say is that we could take C to be any constant meaning any real number and then why would still satisfy this differential equation? We want to find a particular solution. So basically, what we&#x27;re going to do is use this point in the general solution and the right, the exact function. Why would satisfies this differential equation and passes through this point? But so let&#x27;s put in Ah, this value off the point in this general solution meaning we put X equals to one and why equals two to overeat. So what we get is two over e equals. See, time&#x27;s you to the negative one. So that e to the negative one. It seemed as one of working. So get over equals to see over e. So that means C equals to two. So our particular solution y p equals two times e to the negative one of Rex

Mike Gaerlan · Answer

All right. So, in. To find the general solution of this equation, let&#x27;s first turn this into differential operator form. So that becomes D squared. Um, and then B squared minus D minus two of why is equal to zero. So then now are corresponding. Auxiliary equation P of our is equal to r squared minus R minus two on. We need to say that able to zero. So if we want to attempt of after this, we take a look at the factors that to which are just one into, um, and look at the difference. So one and two is going to be negative one. So that matches this here. So then we can talk to this into our minus two and r plus one single zero. So our roots are negative one and two. So our general solution Why of X is going to be able to see one e to the negative X plus C two, e two, the two X right. So this is our general solution here.

Mike Gaerlan · Answer

to find the general solution of this differential equation. Let&#x27;s start by finding the this in differential forms. That&#x27;s gonna be plus t swear. Plus two D plus two of y +00 So then our corresponding auxiliary equation p of our is equal to and then we have our square plus to R plus two is equal to zero. So, uh, let&#x27;s try to factor this. So that means we need to take a look at the factors of to that that add up to two. So the factors have to are only one and two, which cannot add up to two. So therefore, we need to use quadratic equation. Just all this. So we&#x27;re gonna have our is equal to negative B, and then we have plus or minus B squared, which is four or square root of B squared, minus four minus four times a C. A s one C is too. And then all divide by two. So in front we have that&#x27;s gonna be a negative one here and then plus or minus four minus four times to. So that becomes four minus eight, four minus eight year. So then that becomes in total negative four square root of negative four is going to be to eye to eye. And then we have this divided by two on the bottom, so that just becomes plus or minus. I hear. So are a here is going to be negative one and R B is just one. So our general solution by of X is going to be equal to C one e to the negative X, then sign of B, which is one X and then plus C two heats of the negative A which is negative X and then co sign of B X, which is just co sign of X. So this is our general solution here.

Problem

Determine whether the given equation is the gener…

Problem 3 Easy Difficulty

Determine whether the given equation is the general solution or a particular solution of the given differential equation.
$$\frac{d y}{d x}+2 x y=0, \quad y=e^{-x^{2}}$$

Answer

Particular solution

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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