20111101, 23:08  #1 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 
Differential Equations Extra Credit
Hello, nerds!
And an acknowledgement that my knowledge of analysis is rather (very) limited, and that certainly R.D. Silverman (and probably many or most of you who look here) know it better than I do. But that's why I'm asking! Extra credit problem on an exam from Monday, with no partial credit. I tore the sheet out and took it with me, mostly due to the partial credit thing. Problem: Code:
Let Y=c1*y1+c2*y2 be a general solution to the equation y''+sin(x)*y=0 . Show that for any consecutive zeros of y1, a and b (a<b), there exists a unique c:[c is an element of (a,b)] such that y2(c)=0. In other words, there is exactly on zero of y2 between any two consecutive zeros of y1. My progress consists only of noting that with reduction of order, y2=c*Integral[1/([y1]^2),x] (in Mathematica notation), and that the Wronskian W = y1*y2'  y2*y1', which is y2*y1' when y1 is zero. (And then neither y2 or y1' is zero by the linear independence of y1 and y2.) Any suggestions? Also, can anybody give a short intro to TeX, maybe calculusspecific TeX? Last fiddled with by Dubslow on 20111101 at 23:10 
20111102, 00:06  #2  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:
is where I bookmarked for TEX just not sure what packages are useful all the time. 

20111102, 02:40  #3 
Dec 2010
Monticello
703_{16} Posts 
As for Tex...just go advanced mode, see the Tex button on the toolbar.
Now, as for the problem, what happens if you sub in C1*y1 + C2*y2 for y in the original ODE. I think you'll find clues there. 
20111102, 11:37  #4  
Nov 2003
2^{2}×5×373 Posts 
Quote:
BTW, the problem/notation is very poorly posed. To begin, a and b are undefined. Neither are c1, and c2 for that matter. Nor y1 and y2. It may be clear from context that a,b,c1,c2 are real numbers. But then, the notation sucks. If one uses c1, c2 as reals, then y1, y2 should also be reals. It should say y1(x) and y2(x) instead. 

20111102, 15:22  #5 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3·29·83 Posts 
You knew what I meant.
Let a, b, c, c1, and c2 be real numbers. y1(x) and y2(x) are linearly independent functions of x that solve the given differential equation. a is a number such that y1(a)=0. b=a+h, where h is the smallest number greater than zero such that y1(b)=0. Then show that there is exactly one c on the interval (a,b) such that y2(c)=0. No one has mentioned Sturm sequences in class, so I don't think it's necessary (otherwise this problem wouldn't have been on a test). @Christenson: To me, that just seems to lead to c1*y1'' + c2*y2'' + sin(x)*c1*y1 + c2*sin(x)*y2=0. Group this as [ c1*y1'' + sin(x)*c1*y1 ] + [ c2*y2'' + sin(x)*c2*y2 ] = 0, which tells us nothing new. Last fiddled with by Dubslow on 20111102 at 15:25 
20111102, 15:39  #6  
Nov 2003
2^{2}·5·373 Posts 
Mathematics is a language in which it is possible to say exactly what is
meant. Learning to do so is part of learning mathematics. Quote:
were posed that were outside of what was taught in class. Besides, Sturm sequences are often taught prior to taking diffeq's. They arise in the context of finding zeros of polynomials. 

20111102, 15:46  #7  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
7221_{10} Posts 
Quote:
I had never heard of them before, and the first extra credit problem (this was the second exam) fell well within the class material boundaries. I will take a closer look though. 

20111105, 16:39  #8  
Apr 2010
2^{2}·37 Posts 
Quote:


20111107, 17:52  #9 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 
Thanks. Did you see that somewhere or did you make it up yourself? (And if so, what's your experience with math?)

20111107, 23:07  #10  
Apr 2010
2^{2}×37 Posts 
Quote:
Taking further into account that every mathematical insight I have ever found myself has regularly turned out to be at least 250 years old, you are bound to find it in some textbook authored about one *The time of Wronski and Abel 

20111108, 00:19  #11 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3·29·83 Posts 
Indeed. We went over Abel's identity about a month ago.

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