Differential Equations Extra Credit

[Dubslow]

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:

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.

The material on the test concerned linear equations, generally second order. Methods covered were undetermined coefficients, variation of parameters, reduction of order, Euler equations, and power series solutions.

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 calculus-specific TeX?

[science_man_88]

Originally Posted by Dubslow

Also, can anybody give a short intro to TeX, maybe calculus-specific TeX?

http://ctan.mirror.rafal.ca/info/sym...symbols-a4.pdf

is where I bookmarked for TEX just not sure what packages are useful all the time.

[Christenson]

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.

[R.D. Silverman]

Originally Posted by Dubslow

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:

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.

The material on the test concerned linear equations, generally second order. Methods covered were undetermined coefficients, variation of parameters, reduction of order, Euler equations, and power series solutions.

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 calculus-specific TeX?

As a first cut at the problem, have you looked at Sturm sequences?????

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.

[Dubslow]

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.

[R.D. Silverman]

Originally Posted by Dubslow

You knew what I meant.

Mathematics is a language in which it is possible to say exactly what is
meant. Learning to do so is part of learning mathematics.

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).

I have taken quite a few math classes where "extra credit" problems
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.

[Dubslow]

Originally Posted by R.D. Silverman

Mathematics is a language in which it is possible to say exactly what is
meant. Learning to do so is part of learning mathematics.

It's rather more difficult here than on paper, especially since I don't know TeX. I did so anyways, at least the second time.

Originally Posted by R.D. Silverman

I have taken quite a few math classes where "extra credit" problems
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.

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.

[ccorn]

Originally Posted by Dubslow

Problem:

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.

1. Show that the Wronskian is constant, i.e. independent of x.
2. Show that y1'(a) and y1'(b) have opposite signs.
3. Use the two facts above to show that y2(a) and y2(b) have opposite signs.
4. Conclude that y2(x) has at least one zero between consecutive zeros of y1(x).
5. Swap the roles of y1 and y2.
6. Conclude that y1(x) has at least one zero between consecutive zeros of y2(x).
7. Optional: Note that sin(x) plays almost no role here; generalize.

[Dubslow]

Thanks. Did you see that somewhere or did you make it up yourself? (And if so, what's your experience with math?)

[ccorn]

Originally Posted by Dubslow

Thanks. Did you see that somewhere or did you make it up yourself? (And if so, what's your experience with math?)

I have not seen it elsewhere, but given my education in the mathematics of applied mechanics, not finding my way to it would have been a shame.

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 generationcentury* after Leibniz.

*The time of Wronski and Abel

[Dubslow]

Indeed. We went over Abel's identity about a month ago.