5

Find the general solution of the differential equation 2y' + ry + (52? _ 9y = 0 _Answer: y = Cnzn n=mn whereif n > m and n = 2k is even,ifn > m and n = 2...

Question

Find the general solution of the differential equation 2y' + ry + (52? _ 9y = 0 _Answer: y = Cnzn n=mn whereif n > m and n = 2k is even,ifn > m and n = 2k + 1 is odd,mand c is an arbitrary constant:Choose your answer so that Cm = 1 Note: Because the coefficient of y" is zero when 1 = 0, you will not fnd two linearly independent solutions of the differential equation defined near € 0. In fact; by looking at the differential equation, You can see that any solution defined on an o

Find the general solution of the differential equation 2y' + ry + (52? _ 9y = 0 _ Answer: y = Cnzn n=mn where if n > m and n = 2k is even, ifn > m and n = 2k + 1 is odd, m and c is an arbitrary constant: Choose your answer so that Cm = 1 Note: Because the coefficient of y" is zero when 1 = 0, you will not fnd two linearly independent solutions of the differential equation defined near € 0. In fact; by looking at the differential equation, You can see that any solution defined on an open interval containing I = 0 must have y(0) = 0. Submit answer



Answers

Determine two linearly independent solutions to the given differential equation of the form $y(x)=e^{r x},$ and thereby determine the general solution to the differential equation. $$y^{\prime \prime}-36 y=0$$

So we need to find solutions to this different situation of the form. Lie of X is equal to either the Rx. So we need to find why Prime, which is going to be able to Ari to the Rx and why Double prime of X is equal to r squared E to the R. X. Next, we substitute these into a differential equation here. So we get r squared E to the R X minus 36. Uh, sorry. 36 e to the R X is equal to zero. Then we can factor out and eat of the Rx from both of these trips. Either. The Rx R squared, minus 36 is equal to zero. This eat of the Rx cannot be equal to zero. So therefore we only need to set this part of the equation equals zero that's R squared minus 36 is equal to zero is equal to zero. So r squared equals 36 so that our is going to be equal to plus or minus six. So six and negative six. Therefore, our two solutions are gonna be equal e the six x either negative six x S o r. General solution. Why Sea of Acts is going to be equal to is able to see one e to the six x plus c two e to the negative six x and that's these. There are solutions.

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

Okay, so this question we're asked to find the original differential equation. And so to do that, let's go ahead and rewrite this A C one E. To the zero X plus c two E two the two X. And in doing this, we're going to be able to identify our values easier. And so let's go ahead and do that. We'll have our equals two, zero and two. So this means are factors were R times R -2 And it's equal to zero. So we can expand this a little bit and doing so well. Have r squared minus two are equals to zero and we can go ahead and replace these are scored in our values with our original substitution, which would have been y double prime -2, y prime equals to zero. So for this question this is going to be original differential equation and or an answer.

So we have the differential equation y double prime plus for why is it zero? And we need to find solutions of the form Wife of X is equal to eating our eggs. So first we need to find, um Why, prime of X which is equal to our eat the aretz. Why double prime of X is equal to r squared either the Rx. Now we plug these into our equation here, so we do our square feet to the R X plus or O. That should be why prime by the way, plus four r e to the R X is equal to zero being factor out either the Rx you, the Rx times are squared us for our people to zero. Now, since even the Rx can never equal zero, then we just need to set off a squared plus four are equal to zero equal to zero. Here we can background and our from both terms it just becomes our times are plus four. So we get either r equals zero r equals zero or negative for and remember, eats of the zero is equal to one. So our two solutions are going to be one and eats of the negative four x right, So okay,


Similar Solved Questions

5 answers
How long will take an Investment double answier decimal places:)the interest rate 6o0 compounded continuously? (Round yourWhat the equlvalent annual Interest rate? (Round vour answer two declmal places:)
How long will take an Investment double answier decimal places:) the interest rate 6o0 compounded continuously? (Round your What the equlvalent annual Interest rate? (Round vour answer two declmal places:)...
5 answers
JdronFind the general solution to: -8' +4ly = 0 Give your Ans1 15 } In your answer_ and €1 denote abutnny corant and , 15 c2 the independent variabke: Enter € 4s cl and c2Answer:Submlt answerPrevous
Jdron Find the general solution to: -8' +4ly = 0 Give your Ans1 15 } In your answer_ and €1 denote abutnny corant and , 15 c2 the independent variabke: Enter € 4s cl and c2 Answer: Submlt answer Prevous...
5 answers
The data givcn follou. Excel File: data14-17.xlsx#Z The estimated regression equation for these data is & = 7.0+.&- Compute SSE, SST; and SSR (to decimal)SSESSTSSRWhat Petcentage of thc tota bumsqudraeaccounted (arthe estimated regression cquation (to decimal)?What the valuethe sample correlation cocfficient (to decimal)?
The data givcn follou. Excel File: data14-17.xlsx #Z The estimated regression equation for these data is & = 7.0+.&- Compute SSE, SST; and SSR (to decimal) SSE SST SSR What Petcentage of thc tota bum squdrae accounted (ar the estimated regression cquation (to decimal)? What the value the sam...
5 answers
PROBLEM 2 [points: 10]Suppose you're given a list of 50 gelato flavors and asked to rank your top 3. How many possible "top 3" lists could be there in total for a person to have?If person doesn specify the order; just gives 3 "favorite" flavors?If person specifies which flavor exactly is herhhis #1,#2,#32
PROBLEM 2 [points: 10] Suppose you're given a list of 50 gelato flavors and asked to rank your top 3. How many possible "top 3" lists could be there in total for a person to have? If person doesn specify the order; just gives 3 "favorite" flavors? If person specifies which f...
5 answers
3 IL ex(x2 (z4 _
3 IL ex(x2 (z4 _...
5 answers
9.) (14) Write a definite integral (do not evaluate) to represents the volume obtained by revolving the region in the first quadrant enclosed by the curves y = | and y=r about they-axis, using each method The disk/washer method,The shell method
9.) (14) Write a definite integral (do not evaluate) to represents the volume obtained by revolving the region in the first quadrant enclosed by the curves y = | and y=r about they-axis, using each method The disk/washer method, The shell method...
5 answers
Part of 4Would De unusual if tne sampl proportion of tax returns for which no tax was paid was ess tnan 0.23? (Choose one) pe unusual if the sampl proportion of tax returns for which no tax was Paid was less than 0.23, since the probability
Part of 4 Would De unusual if tne sampl proportion of tax returns for which no tax was paid was ess tnan 0.23? (Choose one) pe unusual if the sampl proportion of tax returns for which no tax was Paid was less than 0.23, since the probability...
1 answers
Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{l} 3 x-6 y \leq 12 \\ -x+2 y \leq 4 \\ x \geq 0, y \geq 0 \end{array} $$
Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{l} 3 x-6 y \leq 12 \\ -x+2 y \leq 4 \\ x \geq 0, y \geq 0 \end{array} $$...
1 answers
In quadrilateral TWAI, $\overline{\mathrm{TA}}$ and $\overline{\mathrm{IW}}$ bisect each other. Does it follow that $T A=1 W ?$
In quadrilateral TWAI, $\overline{\mathrm{TA}}$ and $\overline{\mathrm{IW}}$ bisect each other. Does it follow that $T A=1 W ?$...
5 answers
0 0 0 What sp3d2 sp2 ds sp3 5 the approximate hybridization state of the oxygen molecule 5 ethanol, LHOSHZ)
0 0 0 What sp3d2 sp2 ds sp3 5 the approximate hybridization state of the oxygen molecule 5 ethanol, LHOSHZ)...
5 answers
With its graph in a–f on the next page$z=x^{2}+y^{2}$
With its graph in a–f on the next page $z=x^{2}+y^{2}$...
5 answers
A 2.5kg sphere with a radius of 35cm rotates on an axis throughits center. A point on the edge of the sphere is measured to have atangential speed of 10m/s. (a) What is the rotation rate of thesphere? (b) What is the rotational kinetic energy of the sphere? Ifthe center of the sphere moves with a translational speed of 3m/s,what is the total energy of the sphere?
A 2.5kg sphere with a radius of 35cm rotates on an axis through its center. A point on the edge of the sphere is measured to have a tangential speed of 10m/s. (a) What is the rotation rate of the sphere? (b) What is the rotational kinetic energy of the sphere? If the center of the sphere moves with ...
5 answers
A huge block of ice was left outside in the form of truncated prism: The base is square with an edge of 12 feet: Two adjacent lateral edges are 20 feet long and the other two are 14 feet long: Find the volume of the ice now and the original volume if the ice if it has already melted 25% of the original since it was left: Lastly find the surface area of the ice:
A huge block of ice was left outside in the form of truncated prism: The base is square with an edge of 12 feet: Two adjacent lateral edges are 20 feet long and the other two are 14 feet long: Find the volume of the ice now and the original volume if the ice if it has already melted 25% of the origi...
5 answers
Nickle(A) 2.034Intensity 1001.762421.246 1.06221201.0170.8810.808140.78815Potassium chlorided (A) 3.150Intensity 1002.224591.816231.5731.407201.284131.1131.0490.9490.9950.9080.8730.841
Nickle (A) 2.034 Intensity 100 1.762 42 1.246 1.062 21 20 1.017 0.881 0.808 14 0.788 15 Potassium chloride d (A) 3.150 Intensity 100 2.224 59 1.816 23 1.573 1.407 20 1.284 13 1.113 1.049 0.949 0.995 0.908 0.873 0.841...
5 answers
Woi 4sHow did Jesus study algebra? What did you find the most interesting and why? How can you apply this article to you and your study of algebra? How do you think the study of mathematics further can help you develop and strengthen your relationship with God?
Woi 4s How did Jesus study algebra? What did you find the most interesting and why? How can you apply this article to you and your study of algebra? How do you think the study of mathematics further can help you develop and strengthen your relationship with God?...
5 answers
Consider the following diagram, in which single lens is used to form an image of an arrow shaped object If the lens is moved to the right will the image move and if so what direction?
Consider the following diagram, in which single lens is used to form an image of an arrow shaped object If the lens is moved to the right will the image move and if so what direction?...

-- 0.069359--