Question
(a) Solve the Euler equation ry" 3rv' + 4y = 0. You should obtain only one solution. (b) We know that there should be two solutions_ Show that r2 lnI, I > 0, solution is also
(a) Solve the Euler equation ry" 3rv' + 4y = 0. You should obtain only one solution. (b) We know that there should be two solutions_ Show that r2 lnI, I > 0, solution is also
Answers
Find the general solution to the given Euler equation. Assume $x>0$ throughout. $$4 x^{2} y^{\prime \prime}+y=0$$
Okay, so here's the problem that's given to us. The first thing that you want to do is isolate the square root so that you can square both sides and get the are out of the square root. So what I'm going to do first is At our and subtract two from both sides. And once I do that I am left with screwed of R plus four equals ar minus two. Now I can square both sides and get that are on the left side of the equation out of the square root. When I do that I have are plus four equals R squared minus four. R plus four. And now what you can do is subtract are from both sides and subtract four from both sides. Once you subtract are from both sides, you'll have four equals R squared minus five R plus four. And then the fours will cancel out from both sides of the equation So that your life was zero equals r squared minus five. Are, what I would do here is like a reverse of the distributed property, which is that you would take our out of the equation are out of Part of the equation so that you have zero equals r times ar minus five. And by doing this you can then set up to equalities. Um because multiplying any Number Times zero gives 0, you can set up this equation so that R equals zero and ar minus five equals zero. Because if either one of those equations happens to equal zero then the other side will too. So from our equal zero and ar minus five equals zero, we get the solutions zero and five. So now we have to plug those back into the initial equation to see if they both still ring true. By plugging zero into the initial equation we would get square root of four minus zero plus two equals zero. Square root of four is obviously just two, and then zero is a non factor, so we have two plus two equals zero, which obviously is not true, Which means that zero is not actually a solution. Meanwhile, if we have our equals five, we can plug that back into the initial equation to have the square root of 5-plus 4 minus five plus two equals zero, and then five plus 4 to 9 in the square root of minus three. So three minus five plus two equals zero Is true because then you get zero equals zero. So here five is the only solution And R. equals zero is not an actual solution.
And we'll be able to find the two values of arses that very close to it is the power Rx is a solution of valuable less managed by this, managed to wage goes to zero so valuable this madness providers manage to constitute so we can write, did to weigh upon D x squared minus of levi upon dx Months of two by the cost of zero. So we can get this carmen is a D -2. Dubai close to zero. Speak and write do you monastery into deep Lisbon because of curiosity cost too. And the coastal minus one So solution can return is very costume into into the power do we express being two to the power one has run into X. So the value of RS two, the value of artists too and minus one sweet and it is going to advise you is we want to from here weekend I did, that is a plus B is because +21 is from here, we can edit A Plus B equals two and avoid a zero. That is because of to A to the power to X. Excellent job minus p into it to the power minus off, Do you know that is the question to twice of a minus or B. That is question too. So by following these two equations So three questions three, he will be close to one, he is one, B is zero. The solution is solution is what it costs to, He is one of the power two weeks place being to be a zero. So so results, bye bye constituted the power works, I hope you understood. Thank you. And the value of artists, too. And one is what.
Alright, Prime number two X squared wide over prime plus X Y prime minus. For a while, I equals zero. Find the journal solution using Ah, the given or other equation. So we'll use our squared plus one ones +10 R minus four equals zero. Do we factor that we get our money? Two times are plus two equals zero So articles to common Negative too. And therefore wise it good too. C one x squared plus you two over.
10 quadratic For our using the quadratic formula, we found that are is a good A one plus or minus. I using this farm. Okay. Why it calls X see one co sign. What next? Let's see to sign.