5

(6pts) Find the general solution to the differential equation: y" +y' 42y = 0....

Question

(6pts) Find the general solution to the differential equation: y" +y' 42y = 0.

(6pts) Find the general solution to the differential equation: y" +y' 42y = 0.



Answers

Solve the differential equation.

$ y'' - 6y' + 9y = 0 $

In this problem, we have to solve the differential equation. Why? Double crime minus y prime minus six. Why is he called to zero so forthis kind of differential equation? We can a specific characteristic in creation. So the characteristic decoration it's gonna look like the square minus T minus six. Is it called a zero? Now let's try to solve this equation. I can see that there is a nice factory ization. This can be factored as t plus two dines. T minus three is a call to zero so that in place either tease a call to negative to Ortiz is a call to positive. Three. Certain The fundamental solution are going to be Ito the poor. Negative two times six on each of the power, three times six. So then, general solution, it's going to look like Why is he going to see one? For some constant? It'll depart negative to x plus C two you to the poor three x So that's going to be might general solution

All right, silver problem. 17. You 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 assume that the solution is in the form of E to the power for constant times the independent variable. We're going to use X for the independent variable, but you can use any variable you want and likewise, we're gonna divide. Derive this. So why prime Miss Movie K Times, Seats of power, Chaos And why don't finally see what's in the case query times if the parking axe and only substitute these into the differential equation. So it's going to be Kase Goro times you to a party next time K time teeth of parking explosive. You need to have heart attacks. Says he wants a zero. We substitute on T to the power tax. So it's gonna be you took architects Times Case carried for us. K post One is he was a zero. We now find their values for K such that this equation equals zero. We know that the left side is never gonna equal to zero since is an exponential function misreadings We're gonna rely on the right side T equals zero and conveniently is just a quadratic equation. So its case growing up last night, case group was came. Plus one supposes Europe and we're going to use the quadratic formula. So negative one poster. Nice a square once, Grace Just one minus four times one size one over two times weren't neither one Folsom lines the spirit of negative 3/2. That's very with omega three. If you was, uh, I have science route three. So in the end, our solutions for K are gonna be negative. One force from Linus. I've rich three over its here, and we just deployed these values into the solution formats. So it's gonna be why, in secret, too, the first constant times e to the power of negative one Linus, I routes 3/2 X and then we add the other constant with it. So it's gonna be plus the second constant times he's a powerful negative one plus for three I over two x And now this curve Yes, I mean this. Generally, we don't consider this a solution, since we have to like, make sure everything is in the real world. We can't have any imaginary values. So before we do, before we go on, we're gonna sports cars. These the turns. So it's going being each of our like that 1/2 x times U to the power of negative. I wrote 32 x and on the right side we're gonna have years and power negative one have X times e to the power of positive I heard 3/2 x and we factor on e to the negative 1/2 excess movie each a negative Excellent tee times See one e t. A power of negative I read three x plus the teeter power constant Iran +32 x And in order to turn thes terms in tow, rial terms working, we would use the McLaurin Siri's for you to the part of acts and sign next and close on xto working out. But it's a lot of writing. So basically the formula and you only through among for us Eton Park I a times a constant will say beta times the independent variable acts If you go to co signed of data X first I times sign of Data X and we're gonna use this formula. It's Ah, make these terms, riel. I guess so. Start with the first term. Using the power. Negative. I threw 32 X It's going to go toe co Sign off, Meg. Never. 3/2 X plus I times sign off. Negative route through with your tax co sign isn't even function, which brings the legs of here is not going to matter whereas the Sinus an odd function which means the negative inside is gonna turn into a negative sign function. And now we do the same for the second term. So it's gonna be e to the positive I threw 32 X Yes, he was co sign of There are three to ax. Plus I Time sign We're 32 X and that we just replaced these terms with these Cosan and sign functions. So we're going to focus on this parts, so it's gonna be see one times co signing off. I'm gonna be substituted for 32 x with C. Since it's gonna be easier to write. So she is going to go toe rule 32 X So it's gonna be co sign of C minus. I times sign with tea for us. C. Two times Kerr. Sign of tea for us. I time sign of team We know foil both of these turns. So it's gonna be See one co sign of tea minus seal on times eyes time sine of t plus See to science coastline tick for a C two times I times sign of C We now combine like terms so the coastline functions and the sine functions So it's gonna be see one for C to of course I'm see Plus negative C one plus e t times I time sign of fake And we're going to simplify this by introducing new Constance. So when you add the two constant numbers together like C one plus C two, well, it's just gonna turn out to be another constant number, which will say C three. Likewise, if you're subtracted to constant numbers together and multiply them by imaginary number, I Well, this whole thing is also gonna turn out to be another constant number, which will say C four and then we substitute this whole thing back here and this will give us our final solution. Actually, So our final solutions can be why secrets at eight and negative x over two times seem stirring, whether number you like. Two years Times Co Sign of through 3/2 x were started shooting t Vanquish North very much. Iraq's and plus C four times Sign of Ruth 3/2 x and, yeah, this is basically adds

Well, even the question on the form. Why number Prime Minister to well, Bram Rico to zero. The first amulet considered characteristic in question. We will have a square months to um people deserve. And we confronted our on site. And we have a problem. Honest you in coach is zero isn't implies that I could use are on our way. But you two therefore we have a social will be under form. Why go dio the constancy one Agent powers 30 on then plus seat you times eating about duty He and uh a djellaba zero th other constant c one plus c to each of actuality. So this guy, the solution here.

All right, So to find this differential equation or didn't find the general solution to this, we're first going Teoh, turn it into, uh, a linear operator. So that's gonna be D cube minus d squared. Plus D ah, minus one of why is it zero? So the corresponding auxiliary equation p of our is going to be equal to our cute minus R squared plus R minus one is equal to zero. So, uhm, I'm going to attempt to factor this by using by using synthetic division. So I write down the factors. One negative, one one negative one here. And I'm going Teoh trying to factor it using the roots of one. So remember, by rational Ruthie Room. Since this is one here, we only need to look at the factors of this in order to get the roots. So I'm going to try one. So if I do one here, so 11 then that becomes 00 term zeros. 111 Ah, so get zero. All right, so then that means I can factor this Ah polynomial here into our minus one. And then these are the coefficients of the remainder. So it's gonna be r squared plus lips R squared plus one here that's equal to zero. So then now with this, I get r is equal to one And then with this here I get r is equal to r R squared X equal to negative one. So I get are is able to plus or minus I. So that means our general solution is gonna be why of X is equal to. So I start with this route first, So see one e to the X and then I do these two routes next. So this is gonna correspond to a plus or minus eso. Since we have an imaginary it's gonna be co sign and sign S O. C. Two and then the real part is zero. So I'm just gonna go straight to sign of X and then plus C ah, three design of X like so. So this is gonna be our total general solution here


Similar Solved Questions

5 answers
3 3.0Bi =403.0 m17. What is the magnitude of the magnetic flux passing through the square loop of conducting wire shown the figure? 18. If the flux changed to 20 Wb over a period of time, what would be the direction of the induced magnetic field in the loop of wire? What would be the direction of the current?
3 3.0 Bi =40 3.0 m 17. What is the magnitude of the magnetic flux passing through the square loop of conducting wire shown the figure? 18. If the flux changed to 20 Wb over a period of time, what would be the direction of the induced magnetic field in the loop of wire? What would be the direction of...
5 answers
Possible, find AB, 84, and 42, (IE not possible enter IMPOSSIBLE: 3| e ["Need Help?[-/1 Points]DETAILSLARPCALC1O 8.2.005Sove[5 2] - [~;Nced ! Holp?
possible, find AB, 84, and 42, (IE not possible enter IMPOSSIBLE: 3| e [" Need Help? [-/1 Points] DETAILS LARPCALC1O 8.2.005 Sove [5 2] - [~; Nced ! Holp?...
5 answers
8.00 tank at - 7,8C k fllcd with 2 4} dinitrogen difhxride pus and [J,8 ot diorine penta luoride gas YOu cn assure both gases behave Ideh gases under these condibons Cakculate thte mole Iraction eoch Qus Round cic Motunstcny spniltcont depits.nnAnitye den dnuonidtchlorlnn nankalltetedn
8.00 tank at - 7,8C k fllcd with 2 4} dinitrogen difhxride pus and [J,8 ot diorine penta luoride gas YOu cn assure both gases behave Ideh gases under these condibons Cakculate thte mole Iraction eoch Qus Round cic Motunstcny spniltcont depits. nn Anitye den dnuonidt chlorlnn nankalltetedn...
5 answers
7.2.8-TQueation Helpcanxinit conlxuureu intonyb Do onaolmt cllotna oistnbuton !ppEesLac lnHnlo UetnnetHaaannPEroPr LEntreIkAalnatn Hmihia cncttonackScact Iho ~oreechalce hothanno necranIntantantcomototaeel(Houndtwo docurulpuicuuneadad |Hocimug nlacotnuatrtd ,Ihnitrannormal dxnhulion nor Inu dibirbua Mepid(Kend
7.2.8-T Queation Help canxinit conlxuureu intonyb Do onaolmt cllotna oistnbuton !ppEes Lac ln Hnlo Uetnnet Haaan nPEroPr L EntreIk Aalnatn Hmihia cncttonack Scact Iho ~oreechalce hothanno necran Intantant comoto taeel (Hound two docurulpuicuu neadad | Hocimug nlacot nuatrtd , Ihnitran normal dxnhuli...
5 answers
2) Consider the following potential U = x' Find the equilibrit poinpoints_ 6) Find the work done by the force ifthe partieles move from What is the force applied t0 the particle at X 4m ,0o X =4m.
2) Consider the following potential U = x' Find the equilibrit poinpoints_ 6) Find the work done by the force ifthe partieles move from What is the force applied t0 the particle at X 4m , 0o X =4m....
5 answers
Problem 18.03of 9ConstarPart AWhat is the current in amperes if 1300 Na ions flow across cell membrane in 3.2 ps The charge on the sodium the same as on an electron, but positive.Express your answer to two significant flgures and include the appropriate unlts:QA0ValueUnitsSubmitRequest Answer
Problem 18.03 of 9 Constar Part A What is the current in amperes if 1300 Na ions flow across cell membrane in 3.2 ps The charge on the sodium the same as on an electron, but positive. Express your answer to two significant flgures and include the appropriate unlts: QA 0 Value Units Submit Request An...
5 answers
A matched case control study was conducted to study the relationship between stroke and oral contraceptive use in women aged 18 to 44 years of age. Within each pair; a case and a control were assessed for exposure to oral contraceptives_ The following data were observed:Case ExposedUnexposedExposed UnexposedControl4455What is the conclusion of the test?
A matched case control study was conducted to study the relationship between stroke and oral contraceptive use in women aged 18 to 44 years of age. Within each pair; a case and a control were assessed for exposure to oral contraceptives_ The following data were observed: Case Exposed Unexposed Expos...
5 answers
Use a method similar to that suggested in Problem 10 to show that the eigenvalue problem in Problem 6 has no negative eigenvalues.
Use a method similar to that suggested in Problem 10 to show that the eigenvalue problem in Problem 6 has no negative eigenvalues....
5 answers
Write polar equation of conic with the focus at the origin and the given data ellipse _ eccentricity 1/4_ directrix r = 5sec 8
Write polar equation of conic with the focus at the origin and the given data ellipse _ eccentricity 1/4_ directrix r = 5sec 8...
1 answers
Perform the indicated operations and/or simplify each expression. $$\left(2 x^{2}-1\right)\left(3 x^{2}\right)+\left(x^{2}+3\right)(4 x)$$
Perform the indicated operations and/or simplify each expression. $$\left(2 x^{2}-1\right)\left(3 x^{2}\right)+\left(x^{2}+3\right)(4 x)$$...
5 answers
TJS Evctviato8 You standard auport nunway with 122 Ibslft? 1 Vi for the so1l intetval V construction confidence Your H %06 (S L) (15) 11 the for bound confidence lower 0066 (SL)
TJS Evctviato8 You standard auport nunway with 122 Ibslft? 1 Vi for the so1l intetval V construction confidence Your H %06 (S L) (15) 1 1 the for bound confidence lower 0066 (SL)...
5 answers
The $x$ -intercept is on which axis?
The $x$ -intercept is on which axis?...
5 answers
Consider an AC power supply with the frequency f = 60 Hzconnected to an inductor (L = 0.2 H), capacitor (C = 2e-05 F) andresistor (R = 40 Ohm) in series. The maximum current in the ACcircuit Imax = 1 A. What is the maximum applied voltage?
Consider an AC power supply with the frequency f = 60 Hz connected to an inductor (L = 0.2 H), capacitor (C = 2e-05 F) and resistor (R = 40 Ohm) in series. The maximum current in the AC circuit Imax = 1 A. What is the maximum applied voltage?...
5 answers
Points) What equation must be satisfied in order to say "f is continuous I = 19"?points) Suppose that f and are continuous functions and that f(19) = 2 Use the following limit to find the value of 9(19).lim 3f (z)[r)]" [ka)] ) =114 [Hint: Use the definition of continuity:]
points) What equation must be satisfied in order to say "f is continuous I = 19"? points) Suppose that f and are continuous functions and that f(19) = 2 Use the following limit to find the value of 9(19). lim 3f (z) [r)]" [ka)] ) =114 [Hint: Use the definition of continuity:]...

-- 0.070619--