5

Equation (3y2 x)dx + (2y3 6xy)dy = 0 has an integrating factor in the form u(x,y) = u(r + J). (a) Show that this equation is not exact(complete)_ (b) Solve the equa...

Question

Equation (3y2 x)dx + (2y3 6xy)dy = 0 has an integrating factor in the form u(x,y) = u(r + J). (a) Show that this equation is not exact(complete)_ (b) Solve the equation by using U .

Equation (3y2 x)dx + (2y3 6xy)dy = 0 has an integrating factor in the form u(x,y) = u(r + J). (a) Show that this equation is not exact(complete)_ (b) Solve the equation by using U .



Answers

Solve the differential equation by the method of integrating factors and by separation of variables, and confirm that the two solutions are the same.
(a) $\frac{d y}{d x}-4 x y=0$
(b) $\frac{d y}{d t}+y=0$

Try to solve these to such problems with the separation of variables and the amount of integrating factors. So first do separation What variables with the 1st 1. And so this is going to give us dy dx Secret to -3 away. So then I have dy over Why is equal to -3 DX integrating both sides here I have the Helena wife is equal to You get three x. Let's see taking a to both sides here I get why is equal to C. E. To the negative three X against that state. First part by suppression of variables. Next this is my p sequel to three and so then have you is equal to to the In a group of three. She goes here to three X. So now I have the already axe of new times, why somebody to three X. Y is equal to zero times 8 to 3 X zero. So integrating both sides here, constant is equal to Into the three x. Y. So not dividing by into three X. Temple sites. Think of why is equal to see into the -3 X. Just the same. Okay. Second problem again, when you do separation of variables first I got T. Y. DT equals to white. Therefore dy over why Is equal to two DT integrating both sides. I get Elena like equals two T plus you. Okay take a nap both sides. I get why is equal to E. To the two T plus C. Here. She calls cE to the to T. That's my separation of variables here. Let's look at the other side now or the method by integrating method of integrating factors. So P is equal to -2. Therefore it is equal to here to the Integral of -2. We'll see you to the -2. X. Sliding that into that. So that's D. Or detox. Let's eat a -2 x. times y. And that's equal to zero Cereal Toast to zero. So then I integrate both sides. Get a constant is equal to It's a -2 x. That's why Divided by eating. They have 2x. to both sides. That ends up flipping that sign there we get like we'll see E. Two the two X. There should be cheers starting with all repair. There should be a team than to hear T. T. T. And we're done.

The problem We have the Q upon DT equals Q upon care. Our integration did you upon Q equals one upon KDT. So integrating both sides, we have Ln two equal when upon Katie plus C. Our GFT becomes good power. D upon K into old parsi. So from the and Q f t equals. See you power. Do you punky? This is the answer.

Hello, everyone. Whatever You're going to start Problem number 20 from the section chapter Review here the U N differential equation is Why don't you guys plus two y dice? Plus why it was two eggs you to the bar minus X so we can write it does the square plus two D best line, he called zero. You'll find a general solution. It is due Pressler those clerical syrup because minus are common. So general solution can between us. See you are you know, the color Why nothing does. I think X in the bottom minus X then began find departed consolation as like why are because you did a bottom line effects and why toe be X? It'll about minus X so we can find w as do don't turned off Why are by two by around us right toward us. But you remember death You didn't bottom. I know thinks eggs you know the bottom by nothing different station is about sex. He doesn't four minus x minus x feudal about my mother being beginning us. You know the bar my last two x Then we can find you. Yeah, because integral by us like toe half affects they were by W. This is half affects. W why are like my enough expiry tillable my Lessig's India works about minus X. They went by it about a minus Truex so you don't remind us Tricks Peterborough minus X It'll be a pretty good cancer. So in Decorah, minus at the square. So by enough to discard it becomes my last two X cube by three. Similarly, because integral, why you weren't ffx by w Why? When is beautiful four minus x f of X is two weeks you know about minus X. Divided by W is why does a w value you know the bottom While last two X it becomes similar later for my districts ever minus X minus Inter cancer From wind Taylor to Alexis Bruin into square by toe Fuller becomes thanks quiet bees at the square. So CF plus everyone past the right toe ritual begetting us see of is this one that is See you are you know about my nothing plus C to eggs either about minor sex Then if I ever plus beware toe Yeah, I want that is to buy three x cube. It is about a minor sex does a square, Indo and here to about minus six. This is the solution there. Stand off a question. Thank you.

Hello. So today we're gonna work on this problem. And so what it is is the second derivative of Y minus two Y, uh, minus two times the first derivative of y plus why is equal to e to the power of two X? So, as you can see, this equation has two parts to it. There's the left side. And then there's the right side. Now that the right side has just a zero like only a zero, this would be a homogeneous differential equation. But because there is, uh, something on the right side, it is not homogeneous in the solution. Will have two parts to it. So the solution since it has two parts, it'll look like this. Why of X is equal to why sea of X, which is the general part of the solution. Plus why P of X, which is a particular part of the solution. So the general part of the solution goes along with the left side of the equation the blue side and ah, particular side. The solution goes with the yellow side. Eso What we're gonna do is first, we're going to solve for the left side part of the solution to do this, we're going to get We're going to derive a complimentary equation to help us find a solution. And so, the way we're going to do this, we're gonna use the variable r, and then we're going to copy the coefficients of the blue side of the problem. So if this first coefficient is one right, then it's going to be one times r squared. And so, as you see, we're gonna use the variable art, and it's gonna go down by degrees. Okay, so next we have minus two. That's a coefficient, right? So minus two. And then this is only our And then here we have a coefficient of one. So will be plus one positive one. We're gonna set this equal to zero, and then we're gonna solve for R and find the root, which in this case, the way we're gonna do this, um, are is going to be equal to one. So there's only one root for this general side of the equation, the blue side. And when there's only one route Ah, there's a fear. Um, that states that when there's only one route the general part of the solution well, look like this is gonna be see one times e to the power of our X. That's an art. Plus the two times x times e to the power of our So that's an art. Now, if there were two routes, uh, it would be R one and r two been this case r equals one. So actually, one times X, it's just x. And so one time sex is just sex. So this will actually be the blue part. Ah, of the solution. So that is this part, right? Nice. So now we can move on to the yellow part of the solution. The particular part of the solution. So we're gonna do this is we're gonna say y p of X is equal to see a times E to the power of K X. So if we go up here, we can see we can copy, uh, C equals one, right? And then K equals two, right, because she is just one and then or C is just one listen more like this, and then K equals two. So why P of X is actually equal to just a times E to the power of two x. Okay, so we need to actually solve for a and to do that, we're gonna find the derivatives of Wipe. So the first derivative of I P is actually going to be too times a times E to the power of two X and the derivative of the second derivative of wipe is actually gonna be four times a times e to the power of two X. So here we have three different things, right. We have YPF X. We have the first derivative of Y p of X, and then we have the second hoops. We're gonna have the second derivative of Y P of X. So there are three things that we got there and what we're going to do with this information is actually going to copy the original format of a problem. And we're gonna substitute in why and the first derivative of y and the second derivative of y. So what that's gonna look like we're just gonna have the second derivative of why which is for a times E to the power of two X and then we're gonna follow this right, we're gonna follow it. So it's negative two times uh, the first derivative which is to a times e to the power of two X And then we're gonna add just why, Right? Uh, we're gonna add y, which is a times E to the power of two X and we're gonna set that all equal to e to the power of two X and we're gonna solve for a And as you can tell, it's for a time. It's a part of two x minus and then this is going to turn into four a times e to the power of two X So it's gonna be four minus four. Ah, so actually these to cancel out and we're left with only this So a is equal to one and we're gonna substitute that in here. So why P of X, right? Sorry, Y p of X is actually equal to since a is one is just equal to e to the power of two X. So this is actually the second part of the equation. And to get the entire solution, wait to get the entire solution. All of this we have to add this and this together. And so the solution will be why of X is equal Thio. See one times e to the X. We'll see two times x times e to the X plus e to the power of two X and that right there will be your solution.


Similar Solved Questions

5 answers
Predict the species from the list below that will be oxidized first if the following mixture of molten salts undergoes electrolysis. (5 pts) Cu2t, Mg2t, Cl , Br , F Fzs) + 2e" 4 2F-(aq) E' = +2.87 V Clzs) + 2e" 7 2Cl-(aq) E' = + 1.36 V Brz(s) + 2e 7 2Br (aq) E' = + 1.09 V Cu?t(aq) + 2e" 5 Cu) E' = + 0.34 V Mg?t (aq) 2e" - > Mg(s) E' = -2.37VAnswer:
Predict the species from the list below that will be oxidized first if the following mixture of molten salts undergoes electrolysis. (5 pts) Cu2t, Mg2t, Cl , Br , F Fzs) + 2e" 4 2F-(aq) E' = +2.87 V Clzs) + 2e" 7 2Cl-(aq) E' = + 1.36 V Brz(s) + 2e 7 2Br (aq) E' = + 1.09 V Cu...
5 answers
Use the quotient rule to find the derivative of the following (3x2 + 4) (5x+5) y = 6x - 5dy dx
Use the quotient rule to find the derivative of the following (3x2 + 4) (5x+5) y = 6x - 5 dy dx...
5 answers
Enter 0.5 moles Fzlg) was found to have + Hzlg) The reaction Question- Enter the number below: 1 2 HF(g) U a Kc value of 112 RICE at 295K table abtetntotheket expression:Question 53 pts0.5 pts
Enter 0.5 moles Fzlg) was found to have + Hzlg) The reaction Question- Enter the number below: 1 2 HF(g) U a Kc value of 112 RICE at 295K table abtetntotheket expression: Question 5 3 pts 0.5 pts...
5 answers
Identify the allylic halidefs)only /Tand0 Tand MIVend IVQuestion 43iat
Identify the allylic halidefs) only / Tand 0 Tand MI Vend IV Question 4 3iat...
5 answers
{Simpllty your 1 0varabon 3 34 5 TalabonDindy ns1 uluuum"Lzed
{Simpllty your 1 0varabon 3 34 5 Talabon Dindy ns 1 uluuum " Lzed...
5 answers
Using Cantor-Schroder-Bernstein Theorem OR otherwise_ prove that the following two sets have the same cardinality: A = (-1,1) and B = (-1,1)U{-1,1} (b) A = (-2,1) and B = (5,0) U (-2,-1)
using Cantor-Schroder-Bernstein Theorem OR otherwise_ prove that the following two sets have the same cardinality: A = (-1,1) and B = (-1,1)U{-1,1} (b) A = (-2,1) and B = (5,0) U (-2,-1)...
5 answers
Find the angle that is completmentary and supplementary to if 0 = 418 54' 36"2. Calculate the length of are that subtends central angle of measure 0 = IOY on a circle of dliameter 90 cm: (Leave your anSWeT aS simplitied exact value in terms o €)
Find the angle that is completmentary and supplementary to if 0 = 418 54' 36" 2. Calculate the length of are that subtends central angle of measure 0 = IOY on a circle of dliameter 90 cm: (Leave your anSWeT aS simplitied exact value in terms o €)...
1 answers
Determine whether the function is even, odd, or neither. $$f(x)=x^{3} \sin x$$
Determine whether the function is even, odd, or neither. $$f(x)=x^{3} \sin x$$...
1 answers
Evaluate the double integral. $ \displaystyle \iint\limits_D (x^2 + 2y)\ dA $, $ D $ is bounded by $ y = x $, $ y = x^3 $, $ x \ge 0 $
Evaluate the double integral. $ \displaystyle \iint\limits_D (x^2 + 2y)\ dA $, $ D $ is bounded by $ y = x $, $ y = x^3 $, $ x \ge 0 $...
5 answers
3. Briefly describe how G proteins send signals into the cell:Igs
3. Briefly describe how G proteins send signals into the cell: Igs...
5 answers
IPr iPr-SiCl iPr OH EtaNMgBr2. BuANtF , HzO1. LiAIHA 2. HzotKMnOa HzSO4OH HzSO41Q3 2.HzoMgBr 2. H3OtPCC _Brz HzoNazCr2Qz HzSO41 |Pa g eOH1D-MgBr 2. H3O"
iPr iPr-SiCl iPr OH EtaN MgBr 2. BuANtF , HzO 1. LiAIHA 2. Hzot KMnOa HzSO4 OH HzSO4 1Q3 2.Hzo MgBr 2. H3Ot PCC _ Brz Hzo NazCr2Qz HzSO4 1 |Pa g e OH 1D-MgBr 2. H3O"...
5 answers
Question 8 (1 point)Which ofthe following I$ a form of reproduction in bacteria?TransformationTransductionNelther of theseBoth oi theseQuestion 9 (1 point) Transformation way combine nciv genes?TrueFalseQuestion 10 (1 point) Ihe ONA uplaku process does not requlre any enery True: False3 Ati~c;Um-slions >J67 /
Question 8 (1 point) Which ofthe following I$ a form of reproduction in bacteria? Transformation Transduction Nelther of these Both oi these Question 9 (1 point) Transformation way combine nciv genes? True False Question 10 (1 point) Ihe ONA uplaku process does not requlre any enery True: False 3 At...
2 answers
Find the characteristic polynomial ofA =_2please enter your answer using the variable t. Deduce the smallest positive real eigenvalue of A_The characteristic polynomial is c(t)The smallest positive real eigenvalue of A is A
Find the characteristic polynomial of A = _2 please enter your answer using the variable t. Deduce the smallest positive real eigenvalue of A_ The characteristic polynomial is c(t) The smallest positive real eigenvalue of A is A...
5 answers
V 0 Select the Does This (Type =tan The sequence The sequence the sequence Question: I {a} pts diverges. and, converge Or diverge? using z as needed,) if necessary; fill in the Find the YECDOXI0 1 complete the of 20 (1 1 choice. complete)8 select and enter your answer(s)
V 0 Select the Does This (Type =tan The sequence The sequence the sequence Question: I {a} pts diverges. and, converge Or diverge? using z as needed,) if necessary; fill in the Find the YECDOXI0 1 complete the of 20 (1 1 choice. complete) 8 select and enter your answer(s)...
5 answers
Each reactim Povide The Reagents f Show all Wrk
each reactim Povide The Reagents f Show all Wrk...

-- 0.068800--