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Find solution of the problem "rr "uu + "z + " = 0 for [ < " < 2 With the boundary condlitions T = Y a6 T = | aHd U = 13 a6 / =2 (Hint...

Question

Find solution of the problem "rr "uu + "z + " = 0 for [ < " < 2 With the boundary condlitions T = Y a6 T = | aHd U = 13 a6 / =2 (Hint; Make the ansatz " Vi+hr"

Find solution of the problem "rr "uu + "z + " = 0 for [ < " < 2 With the boundary condlitions T = Y a6 T = | aHd U = 13 a6 / =2 (Hint; Make the ansatz " Vi+hr"



Answers

Solve the boundary-value problem, if possible.
$$
y^{\prime \prime}+16 y=0, \quad y(0)=-3, \quad y(\pi / 8)=2
$$

What about the second on the L G A y double Prime plus B y prime? Plus why coaches there are. Then we can define the characteristic question I ask where plus B plus C equal to zero. Then we suppose that we have the two complex root want to equal to the and five plus and minus that I and then we have a general solution Y e go to the C one Ebola and for X times go side off the better X plus C to each other. And for X times I often a bit the ax in this question were given Why number Bram minus eight white Bram plus 17 y equal to zero with the initial value wiser, equal to three. And why on the pie equal to two? Yeah, the first time we need to write down the country's characteristic question. So have the out square minus H R plus 17 equal to zero. Using the quality formula, we have our one to equal to eight plus or minus square root of the 64 minus 68. On with you, they will get equal with you the four plus and minus. Uh, we have here will be, uh I hear. So here we can identify. This could be the un far. This will be the better. And then we have the solution. Why? We go to, you know, see one you know, for X and then go side X plus C Joe Egypt for X Hm? Sai after the X now, using the first initial value here began the wiser or ego to So we will inside this will be one. This will be zero. Therefore, we have equal to just see one on the go to three. Why I am the pie. It wouldn't it wouldn't buy inside cause I'm the side hysterical to the zero, Andi. Then there's only going to see one. So it would be minus say one aged above pie and equal to two. So there's no way so, uh, impossible to find a c two. The Jew. So we have in this question here, we can only find a general solution here only

For this one we got a function our Squire plus for our Plus 13 equals to zero. So our last two to the power of two plus ni ecos to zero. We got our plus to Squire ecos to 99. So our last two ecosystem 90 or positive three eyes are echoes in 192 three. I then by the general solution we have y equals two. Call signs three x signs three X. Done by the initial condition likes equals zero, Y equals two E. Zero. That is 1000 0. That is one son zero ico suit zero. So when ax equals two pi over two, y equals to one. Like you two times pi over two. So next you pie. See one call sign Uh three Pi over two. Mhm. Mhm. Yeah. C two Science three Pi over two. So we can see That's what he calls to 91 and this one jersey calls to zero. So we got one iCO soon not to U C. Two E. Two power of 90 pipe. So see to hear just equal to negative E. To the power of pie. Uh huh.

For this one we got our Squire minus six hour plus 25 week holds +20. So AR -3 Squire Plus 16,000. So we got AR -3 Squire equals 2 1916. AR -3 equals a positive. Thank you for I Are just echoes to three plus or not you for I. So we got a general solution. Mhm. So we got to initial condition that is when X equals to zero, Y equals to one. Coulson zero equals to one. Send zero equals zero. So see one dress equals to one and when actually called soup I. Y. Equals to two, closing for pi equals to one and sign for pi equals to zero. Here we go. See one nickels to to our YouTube Taiwan three pies. So we can see see um got two different values so we can solve this crime.

All right. So for problem 41 had to find a particular social to this differential equation. So when you have a simple differential equation like when all the coefficients are constants, then you consume that. The solution is in the form of each of the perfect constant times the independent variable, which will use x four. So we just arrived. So why I'm crying secrets, Okay, Times either The chaos. Why double front? Because of Kase Goro Times each of the king acts and we just a substitute thes into our differential equation. So it's gonna be in case Kory times either. CAC's I understand science Kids Times easily connects first for thanks, T j KXAS unit zero and we factor on e to the CAC So it's gonna be each of the next Times. Case Group was 10-K for austerity. Flory Secret zero. We know that each of the K X is never gonna equal to zero since it's an exponential function, which means we're gonna have to rely on the right side toe equal to zero and conveniently is just a quadratic equation. Case group was 10-K plus 34 you think was a zero and it doesn't seem to be fact herbal, so it just use the quadratic formula. So it's gonna be negative. 10 plus or minus the square it tens girl minus four times one science 30 floor over two times one which will give us negatives on further minus 100 minus 136 over it. So, *** 10 faltering line is it's grew in 1936 over to negative 36 its secrets Ah, six I or closer minus six I. But we already have a plus or minus. So it doesn't matter. So India and it's gonna be negative. 11th remind us. Six I over Teoh and me simplify self negative five foot minus three I and these will be the solutions to our K. Now there's a more detailed way to solve this in some of the other videos, but we're just going to use a short cut. So I cut down on time. So when when the when the soldiers were K is in the form of off course from Linus beta times, I you can directly reach the solution for the differential equation that goes as follows e to the Alpha times x times a constant times the coastline of Veda X for us and another constant time sign of green attacks. And we just deployed in values. So we have a negative five of post minus three I so our first negative five and Vegas three. So we just put them in. So our general solution to the differential equations gig Why is he goto Eat the things of five Next times steer Juan Cyrus co sign it three x plus C two times Sign it. Three acts And now we just supplicant values to So for the two unknown Constance. So in the lines of zero when x zero Why is this sex? So why is gonna be six is you goto u to the power negative five times zero is just gonna be zero time to see a one Times co sign up three times zero is zero for a seater time sign off zero each of the zeros in the A one so I can ignore that and sign of zero is gonna be zero so we can ignore all of us. And Costa zero is the one striking in order that and the C one Secrets of sex. And now, when excess you goto pie. Why a secretive too? So thinking seriously, goto even negative five high times a C one, which is six times co sign off three pie Forest Theatre times. Sign of very fine. We'll sign off any multiple of any integer multiple of time. There's always gonna be zero So sign of kind Zero sign of two pious zero and those the sign of three plants All those zeros making noise. And you just really immediately from here We know this up for bulls of boundary turns, the C two got canceled out, which brings were never able to sell for C two, Which means of this differential equation with the given boundary values, Yes, unsolvable brawl. And yeah, that's basically s


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