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Merve S.

Differential Equations

6 days, 21 hours ago

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Could you help me to solve it?

mhm. Okay, so for this problem we're just gonna so we have x squared y double prime minus four. My prime minus 14 Y uh four X Y prime minus 14. Y Is equal to X to the 6th. So first we're gonna just solve the homogeneous problem. So the homogeneous problem is when this is equal to zero and the way we're gonna do this is by looking for what solutions of the form Y equals, teach our Y equals X to the er So let's do this. So if you have this then why prime is equal to Rx to the R -1. Why double prime is equal to R times R minus one? Okay. To the R -2. So let's plug it in and then we get X squared Times R Times R -1 Next to the our last two minus four X. Are to the X to the R -1 uh minus 14 X. To the art. I want this to equal zero. So note that we can simplify the excess. So we get Our times are -1 -4 are minus 14, X. to the R is equal to zero. So in particular this implies that so it's uh r squared minus r minus four. R minus 14 is equal to zero. This holds for every X. So this is the quadratic equation -5. R -14 is equal to zero. Now what is our? Well we can solve and this is we can factor this out actually, as are minus seven times are plus two but 14 Plus two Physical 0. So R is equal to seven for Uh -2 are both solutions and -2 are both solutions. So now we have two linearly independent solutions. So we can say that why I can be of the form C one uh X to the seven. Yeah Plus c. two x to the minus uh two. So the so this solves the homogeneous problem. Yeah. And in order to solve the particular problem. Uh huh. So this is the Y home. Now we want to find a particular solution which is one such that X square Y double prime uh minus four. X Y. Prime minus 14. Y. Is equal to X to the sixth. It's hard to find a particular solution. The easiest way is to look for something of the form Y equals X to the sixth or C. X to the sixth. And if you do that, what happens? Well uh so we get again a similar thing. So we got ah 30 x to the 6th Uh 24. Extra six minus R. Okay minus 14 for that seeds. So let me put the seeds as well. So see extra sixth. See extra six, I was 14. C. X to the 6th is equal to X to the 6th. So again this is the same thing in that Well, extra six all cancel out. So we're left with now, 30 minus 24 minus 14 is minus minus eight. C is equal to one Which implies C is equal to -1/8. Right? So this is a particular solution. So particular solution is -1/8 X to the 6th. So this tells us that a general solution is why is equal to the homogeneous solution which is C one extra seven plus C two extra mines too. And then you have a particular solution 18 X to the 6th. And now to calculate what C one and C two are, we use the boundary conditions we have so we have one of, one is equal to eight and y prime of one is equal to minus one. So let's solve this. So the why the first equation tells us eight is equal to C one plus C two minus 18. And the second one tells us uh -1 is equal to 71 Um -2 seats C two and then it's equal to And -68. Okay so let's simplify the system a bit. So we get 98 is able to see one plus C two and for the other one we get uh 3/4. So this will be yeah -1/4 -1 Hour, four is equal to seven C 1 -2 C two. So this is the system we need to solve. And in order to do this let's multiply this one By two and add it. If you modify the two and then add these then what do we get? We got 9 4 -1 4th. It's too is equal to to see one. Uh huh. Yeah to see one plus 71 is 9 C1 and then to see two minus two, C two is equal to zero. So this tells us C one is equal to 2/9. And now let's plug that in. That tells US C two is equal to from the first equation. C two is equal to 9 8 -2/9. Which in our case okay so this becomes 81/8 times nine -16/8 times nine. So this becomes uh 65 over 72. So those are What see one I see to have to be. And then the actual solution is other form this. Yeah. Mhm.

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