01-14-2013, 10:45 PM | #1 |
XFD
Join Date: Mar 2008
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[differential equations] stuck on a problem
Hey this problem seems fairly easy but I keep seeming to mess it up... I'm probably just missing something really obvious.
'Verify by substitution that each given function is a solution of the given differential equation.' (just involves basic derivatives really) y'' - 2y' + 2y = 0 y1 = (e^x)(cosx) y2 = (e^x)(sinx) All this involves is product rule but for some reason I keep messing it up Also, I've been having difficulty with solving systems of linear equations with elementary row operations so if someone could do an example of one for me that would be great. I'll give this system of linear equations. x1 - 6x2 = 5 x2 - 4x3 + x4 = 0 -x1 + 6x2 + x3 + 5x4 = 3 -x2 +5x3 + 4x4 = 0 note: all of the numbers after the x's are supposed to be subscripts. I'll give 10,000 credits to the first person who responds with a correct answer for each problem. Last edited by iCeCuBEz v2; 01-14-2013 at 11:25 PM.. |
01-14-2013, 11:26 PM | #2 |
smoke wheat hail satin
Join Date: Sep 2006
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Re: [differential equations] stuck on a problem
Dusted off some cobwebs in my brain for this one as I'm about 4 years removed from university differential equations, but I have a solution for your first problem. I can work on your second problem in a bit, but the matrix math is a little more tricky than derivatives so it'll take me some time (and again, it's been awhile).
y'' - 2y' + 2y = 0 So let's just start by finding the first and second derivatives. y = e^x*cosx First derivative is as you said - just a simple product rule. Derivative of e^x is just e^x. Derivative of cosx is -sinx. Product rule states (f*g)' = f'*g + f*g'. Therefore, y' = e^x*cosx - e^x*sinx. Agree? Second derivative will follow the same convention, you just have to do it twice. Derivative of -e^x is -e^x. Derivative of sinx is cosx. Following the product rule on each term in y', I get (before simplification): y'' = (e^x*cosx - e^x*sinx) + (-e^x*sinx - e^x*cosx) Which simplifies to: y'' = -2e^x*sinx Putting it all together in y'' - 2y' + 2y: -2e^x*sinx - 2*(e^x*cosx - e^x*sinx) + 2*(e^x*cosx) Multiply yo' shit out and you get: -2e^x*sinx - 2e^x*cosx + 2e^x*sinx + 2e^x*cosx = 0 All of your terms cancel, therefore 0 = 0. |
01-15-2013, 12:00 AM | #3 |
smoke wheat hail satin
Join Date: Sep 2006
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Re: [differential equations] stuck on a problem
Working on the second problem.
Last edited by foilman8805; 01-15-2013 at 12:03 AM.. |
01-15-2013, 12:09 AM | #4 |
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Re: [differential equations] stuck on a problem
The second one doesn't have a solution. Did you type it correctly?
For reference: http://www.wolframalpha.com/input/?i...%2C4%2C0%5D%5D Note the last row of the rref form has 0,0,0,0,1 as the last row. Last edited by dag12; 01-15-2013 at 12:13 AM.. |
01-15-2013, 12:13 AM | #5 |
smoke wheat hail satin
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Re: [differential equations] stuck on a problem
Ninja'd. I just started working it out and I came to the same conclusion.
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01-15-2013, 12:15 AM | #6 |
stepmania archaeologist
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Re: [differential equations] stuck on a problem
For the linear equation one, first we set it up as a matrix equation of the form A*x=b:
Code:
[ 1 -6 0 0 ] [x1] [5] [ 0 1 -4 1 ] [x2] [0] [ -1 6 1 5 ] * [x3] = [3] [ 0 -1 5 4 ] [x4] [0] - Find a row with the first variable, then delete some multiple of that row from every other row with that variable in it, so the first variable isn't in any other rows. Make sure to delete from both A and b, but not x. - Find a row with the second variable and without the first variable, then delete some multiple of that row from every other row, so the second variable isn't in any other rows. - Continue like that (always finding a row with the next variable, but with none of the variables you already did). Ideally you will end up with only one non-0 number in A in each row. However, if you are able to make one row of A completely zeroed, your equation has either zero solutions or multiple solutions. So for this example, let's first eliminate x1 in all rows except the first, by adding the first row to the third row: Code:
[ 1 -6 0 0 ] [x1] [5] [ 0 1 -4 1 ] [x2] [0] [ 0 0 1 5 ] * [x3] = [8] [ 0 -1 5 4 ] [x4] [0] Code:
[ 1 0 -24 6 ] [x1] [5] [ 0 1 -4 1 ] [x2] [0] [ 0 0 1 5 ] * [x3] = [8] [ 0 0 1 5 ] [x4] [0] Code:
[ 1 0 0 126] [x1] [197] [ 0 1 0 21 ] [x2] [ 32] [ 0 0 1 5 ] * [x3] = [ 8] [ 0 0 0 0 ] [x4] [ -8] HOWEVER, if the numbers had been a little different, we might have ended up with something like this: Code:
[ 1 0 0 0 ] [x1] [5] [ 0 1 0 0 ] [x2] [6] [ 0 0 1 0 ] * [x3] = [3] [ 0 0 0 1 ] [x4] [7]
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01-15-2013, 12:22 AM | #7 |
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Re: [differential equations] stuck on a problem
I'm assuming you made up the problem or something just to see how to go through solving a system of linear equations. The particular problem you posed has no solution, but qqwerf's post regarding gaussian elimination should be helpful regardless. Nice post - very detailed, haha (unlike my own post)!
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01-15-2013, 09:37 AM | #8 |
XFD
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Re: [differential equations] stuck on a problem
nope I typed the problem correctly I just picked one out of the book and it said determine if the system is consistent so it is ok that there is no solution. I just wanted to see the row operations.
thank you qqwref and foilman :3 I understand all three of the basic row operations but for some reason the answers I get don't match the back of the book so I'm obviously doing something wrong I just don't know what. Last edited by iCeCuBEz v2; 01-15-2013 at 09:42 AM.. |
01-15-2013, 11:20 AM | #9 |
stepmania archaeologist
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Re: [differential equations] stuck on a problem
Maybe you're making mistakes in the arithmetic? I find that it's often really hard to keep track of stuff like -5 - 2*-1, so it could be that you're doing all the hard stuff right. And you can always check stuff with wolframalpha, like dag12 mentioned (you can even put in only 2-3 rows and see what it does with those).
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01-15-2013, 11:46 AM | #10 |
XFD
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Re: [differential equations] stuck on a problem
yeah probably
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01-24-2013, 09:15 PM | #11 |
XFD
Join Date: Mar 2008
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Re: [differential equations] stuck on a problem
97 on my quiz in linear algebra what up XD
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01-24-2013, 09:16 PM | #12 |
End of the road
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Re: [differential equations] stuck on a problem
u go girl!!!!
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01-25-2013, 02:21 AM | #13 |
smoke wheat hail satin
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Re: [differential equations] stuck on a problem
good shit, bro
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