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Question 34 ptsFind the general solution to Lhe following equalion y" +y 2y = Btet + 10 sin tPlease upload your complele work to GradescopePlease also answer t...

Question

Question 34 ptsFind the general solution to Lhe following equalion y" +y 2y = Btet + 10 sin tPlease upload your complele work to GradescopePlease also answer the following questions about the equation:What is the order of the equation? Select ]2 Is this equation linear? Select ]3. Is this equation homogeneous? SelectCan we apply Ihe method of undetermined coeflicients? Select ]

Question 3 4 pts Find the general solution to Lhe following equalion y" +y 2y = Btet + 10 sin t Please upload your complele work to Gradescope Please also answer the following questions about the equation: What is the order of the equation? Select ] 2 Is this equation linear? Select ] 3. Is this equation homogeneous? Select Can we apply Ihe method of undetermined coeflicients? Select ]



Answers

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation.

$$y^{\prime \prime}-3 y+2 y=\cos (t)$$

So let's start this problem out by substituting the white collar crime and lifetime by prime turns we'll have R squared minus three R equals to zero. And we factory get our tents are months, three equals zero. And that gives us our values of zero and three. And so with that we can build a Hamas in a solution. It's going to be C one plus C two either three X. And we can guess our particular solution which is going to be a chosen two X plus B. Sign two X. So let's take a jury to this twice. So I have Y prime equally negative to a sorry, sign two X plus to be co sign two X and Y double crime. It's going to be negative for a co sign two X minus or B. Sign two X. So we can build our particular sedition or rather solve for it. This is the original equation the wife to plug in our values into. So Y double prime. Yes, negative for a co signed two X minus four B. Signed two X. And then we're going to multiply that by negative three times negative to a sign to X. Plus to be coz I. T. Rex. And that all equals signed two X. And so if we multiply this negative three we end up with negative for a co signed two x minus four be signed two X plus six. A sign to x minus six. Uh huh coz I two X. Yeah. And now we can create a system of equations out of this. So we'll have negative four. Hey um minus six P. Equals to zero. And negative for actually gonna right the other way around six A minus four B equals to one. And so let's multiply the top by six and the bottom by four. So you end up with negative 24 A minus 36 B equals to zero and 24 A minus 16 B. Mhm. Because his sorry not zero close to four. Mhm. Now we can add the terms so that this will cancel out and you'll have 52 negative 52 the equals to four. So he's going to be negative 4 52. Which we can simplify, simplify down the negative 1/13. And so with that we can solve for A. So let's use the equation negative for A minus six P equals zero, negative for a minus six B. Of course to zero. And what's substitute in B. So you have negative for a minus six times negative 1 13 equals to zero. And if we move the term to the other side we get negative for A equals two negative 6/13 and A equals two. Sorry? Yes. Okay A equals two positive three divided by 26. And so our total solution is going to equal are homogeneous solution. Plus our particular solution, in this case we set our particular solution was um C one plus C two. Either the three X. And our particular solution is going to be 3 26. Co sign two X minus 1/13. Sign two X.

Hello. The question is taken from differential equation and where we need to solve this differential equation or we need to evaluate the general solution of this first total linear differential equation. It's a linear differential equation is diva over the X minus VIII. Corsican tax is equal to sign off the wigs. Okay so the situation resembled with the normal differential equation of kind divide over the XP fx and to Y. Is equal to Q. Affects. Okay so how we can solve this kind of equation by evaluating the integrated factor and that integrated factor is integration into the P. X. Dx. If we compare these two equations Pxs minus corsican tax. So integrated factor eventually into the power minus corsican tax D. X. And that will be equal to E. To the power look to the base E. Well she can text plus Kotex since the basis same. So we get this is equal to close second tax plus court of facts. That is the required value of integrated factor. So let us multiply this integrated factor to all the equations we get. Sorry? Music in tax place cortex. Okay into diva. Over the eggs minus corsican tax into cozy can tax plus cortex into why Is equal to sign two weeks into cozy can tax plus cortex. Okay so left hand side of the equation become do you wear the eggs buy into cozy? Can tax plus cortex and right hand side become a sign of to access to sign expo sex and music and taxes one by sine of X. Two Cynics will begin so loud we get to also fax plus. Yes and cortex school sex over Synnex. So cynics will be cancelled out. So we get to course square X. So taking the extra right hand side we get. Do you know why Kosik in tax plus cortex integration and integrating and that is equal to two. Also fax plus oh six can vary journalists who square X. One plus goes two weeks divided by two into D. X. Plus. He sees the constant of integration. So left hand side become why In to go 2nd. Tax plus code affects that is equal to two into integration of host of X. Is equal to sine effects Plus integration of one x 2 is one x by two. And integration of course of two Axis signed two weeks divided by four. Let's see. Okay so taking this course you can takes place cortex to the right hand side. So Jose can tax plus Kotex is one plus co sex over Synnex. So why will be taking to inside wicked Synnex divided by cynics to shine X. So we. Mhm. Soldiers taking calcium also to cynics to to the fore. So they become too for full cynics bless four X. Because two will begin a lot. But this four will come here 1/2 that can fall say next plus two To action to Act two is 4 weeks. And that will be signed to expire too. So that will be sign off. What two weeks Divided by two is in the calcium one second. Let me check this once again. So first I will take the calcium and then do other things. So we wrote that. I can try to terrorists one plus also fax divided by sign effects is equal to two into 4. Cynics plus two wigs plus sign two weeks. Let's see. Divided by for Okay. And now this become why is equal to this will be cancelled out. So we get four signs two weeks plus two weeks plus. So that is not signed to expert. It is sign effects. Yeah. Sign the works. Do I did buy two into 1 plus. Cool sex and signed. Two weeks will go sign of X. Will go here. Let's see. Cynics Divided by one plus schools or facts. Taking this plus 1/2. So this is a required solution of immigration. Hope this clears your doubt and thank

Alright suffer from 11. We have to find the general solution to this differential equation. And when you have a simple differential equation like when all the coefficients are constants, then you can imagine that the solution is going to be in the form of E to the power for constant times, the independent variable. And I'm just gonna put X for the independent variable here. So we're gonna match in. That's of the solution that is going to be in this form. So now we dressed the differentiate. So why prime? That's gonna be K time. Thio Thio Park, Ajax Why double constant being K squared southeast of parquet acts. And now we substitute these into the into the differential equation we want to solve. So it's gonna be a K squared Planets eating to the car K x minus of three times k times easier dark a X minus, 10 times easier park Ajax goes to zero. We're going to factor out a k e to the power k X from all the terms. So it's gonna be each of the parquet acts times que score and minus three K minus 10 and see what his hero and Now we have toe. Make sure like when this part is gonna be equal to zero, since you haven't exponential function on the left, which will never equal to zero. So we can only rely on the second part being equal to zero and conveniently is Jessica for drug equation. So it's gonna be case growing on this three K minus 10 if he was zero, we're going to factor is, since this fact herbal, we're just going to give us K minus five times que my okay plus, which will give us a secure your secret and negative on five. And just like that, we found the solutions to our differential equation, which is gonna be lying sequel to a constant one times E to the power of negative two x plus. We're like, add the two together. So it's gonna be another constant times eat of car fire necks and yeah, that's basically adds

Okay, so let's go ahead and do our general substitution which is going to be replacing. Why? Double prime with R squared and why with one? So in this case will have three hours squared plus one. He calls zero. I can bring the one to the other side or the right hand side. Rather we'll have three. R squared equals negative one can divide by three. So have R squared equals two negative one third. You can take the square root. So have our equals to plus or minus root negative 1/3. So I can rewrite this as our equals to plus or -1 Of Route 1 3rd. Right. And so with this actually we can build our total solution. There are a total solution in this case it's going to be Y equals to see one co sign route one third. X plus C to sign of one divided by three uh squared of that times X. We can sympathize out a little bit. So if we divide uh or distribute the square roots, the numerator and the denominator. In this case The square root of one is simply just one. And so we can go ahead and rewrite this to be one divided by Route three X. Okay. See to sign of one divided by worth three X. And so that will be your answer.


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