determine-whether-the-given-equation-is-the-general-solution-or-a-particular-solution-of-the-give-12

Question

Determine whether the given equation is the general solution or a particular solution of the given differential equation.
$$\frac{d^{2} y}{d x^{2}}+4 y=8, \quad y=c \sin 2 x+3 \cos 2 x+2$$

Vikash Ranjan · Accepted Answer

in the problem we have been given Martinez Last four White L equals eight. And the solution is also given. So so listen given is why equal see sign two X. Last three cost two X plus two. So we have wine ass that is equal to to see Sign two X. So this is cost. So this is cause two weeks minus six sign books. Or we have wide our last that equals -4 seats. Sign to eggs minus 12 cost two X. So while putting the values of white ass and why we have these equals two minus four C. Sign two X minus 12 cars two X plus four C. Sign two x plus 12 cause two X plus eight. This cancer is this part this cancer is this part so we have eight and that is equal to colleges. So this is a general. So listen. Hence it is the answer.

Mike Gaerlan · Answer

to find the general solution. We first set this into differential operated form. So that&#x27;s gonna be D cube minus to D squared, minus four D plus eight of why is equal to zero here like that. And then now we confined the corresponding auxiliary creation, which is going p of our is going to be able to our cube minus two r squared minus for our plus eight. And we said that equal to zero now, uh, an attempt to find the roots. Let&#x27;s try to factor this differential equation here. So to do so, I&#x27;m going to try to use synthetic division to find the roots of this. So I notice here that these are all just multiplied by two. So I&#x27;m going to guess that we have a root of two here. So if I do that, um, actually, let&#x27;s make it negative to instead because if I If I do multiply this by two and this is to this become zero Ah, which is actually, Yeah, I think that&#x27;s what we want. So let&#x27;s try that. So one times two is gonna be too. And then we get zero. This is zero, and then we have negative for down here. Then the two times negative four becomes negative. Eight, which is perfect, hasn&#x27;t zero. So what we have is we have a root of two. So that means we can factor this into our minus two. And then this is the coefficients of our remainder. Recently, B R squared minus four R squared minus four is a with a zero. That means we have a root of two and then our is equal to R squared is equal to four s. So that means our roots are plus or minus two. So we have, ah, to with a multiplicity of two and then also a root of four course, right, minus two. So our general solution, it&#x27;s gonna be why of X sequel to. So let&#x27;s start with our C one and let&#x27;s start with our negative two X and then plus C two, e two, the two X and then are last solution is gonna B plus C three and then x E to the two X for the second multiplicity of the two X. So this are going to is going to be our final general solution here

Amy Jiang · Answer

there&#x27;s so if we saw for our generous solution using online calculator, we get why people to see one fine of X. Let&#x27;s see two closing of X that&#x27;s two ex coast and rex plus one over to x sign of X. Well, given that why, um so is you Go to H. And what a crime of general as you could. Four number four. Okay, so plug me in. What? We know. You know why, Joe, if you go to see to stroll, plus zero is he could, you know. So we have the future. You could be up. You mean we have Why prime you&#x27;re not going to see one plus two is activated for we know that she won is therefore equal to negative six You are solution is wise in containing of six sign of X. It was two ex co sign of X. That&#x27;s one of the two x sign of X

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

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.

Mike Gaerlan · Answer

All right. So to find the general solution of this differential equation, we need to first find the corresponding auxiliary equation. So this is just going to be our plus two square. Ah. So that equal to zero said that to zero. So this is already factored. So then we&#x27;re going to get solutions are is able to negative two and then with multiplicity too. So then our corresponding general solution is going to be Why of X is equal to then we have C one e to the negative two x first, and then our multiple city to means that our next part of the solution is going to be X e to the negative two x.

Problem

Show that each function $y=f(x)$ is a solution of…

Problem 6 Easy Difficulty

Determine whether the given equation is the general solution or a particular solution of the given differential equation.
$$\frac{d^{2} y}{d x^{2}}+4 y=8, \quad y=c \sin 2 x+3 \cos 2 x+2$$

Answer

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