- #1
samleemc
- 9
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if y''-2y'+y=ky k=eigenvalue, y(0,pi)=0, 0<x<pi
find corresponding eigenvalues and eigenfunctions.
thx a lot!
find corresponding eigenvalues and eigenfunctions.
thx a lot!
The characteristic equation for y"- 2y'+ (1- k)y= 0 is [itex]r^2- 2r+ 1-k= 0[/itex] which is the same as [itex]r^2- 2r+ 1= (r- 1)^2= k[/itex] and has roots [itex]r= 1\pm \sqrt{k}= 1\pm m[/itex] with your choice of m as [itex]\sqrt{k}[/itex].samleemc said:if y''-2y'+y=ky k=eigenvalue, y(0,pi)=0, 0<x<pi
find corresponding eigenvalues and eigenfunctions.
thx a lot!
HallsofIvy said:The characteristic equation for y"- 2y'+ (1- k)y= 0 is [itex]r^2- 2r+ 1-k= 0[/itex] which is the same as [itex]r^2- 2r+ 1= (r- 1)^2= k[/itex] and has roots [itex]r= 1\pm \sqrt{k}= 1\pm m[/itex] with your choice of m as [itex]\sqrt{k}[/itex].
The general solution is [itex]y= Ae^{(1+m)t}+ Be^{(1-m)t}[/itex]
Setting that equal to 0 at x= 0 and [itex]\pi[/itex], we find that as long as 1+ m and 1- m are real, A and B must be 0.
In order for k to be an eigenvalue, k will have to be negative so that m is imaginary. Given that, and writing m= ni, [itex]y= Ae^{(1+ni)t}+ Be^{(1-nit)}= e^t(Ae^{nit}+ Be^{-nit})[/itex]. We can write that as [itex]y= e^t(C cos(nt)+ D sin(nt))[/itex].
Now we have [itex]y(0)= e^0(Ccos(0)+ D sin(0))= C= 0[/itex] and [itex]y(\pi)= e^{\pi}(Ccos(n\pi)+ Bsin(n\pi)= Be^{\pi}sin(n\pi)[/itex] (because C= 0) and that must be equal to 0. That will be true either for B= 0 or for [itex]sin(n\pi)= 0[/itex] which will be the case as long as n is an integer.
Can you find the eigenvalues and eigenvectors from there?
samleemc said:k have to be negative and n have to be integer, do u mean k=0 ?!
Please answer ! Thanks !
An eigenfunction is a mathematical function that, when multiplied by a constant, produces a new function that is proportional to the original function.
Eigenfunctions are used in many areas of science, including physics, chemistry, and engineering. They are particularly useful in solving differential equations and representing physical systems.
Yes, there are many resources available to help with understanding eigenfunctions, such as textbooks, online tutorials, and academic courses. Additionally, seeking help from a math or science tutor or consulting with a colleague can also be beneficial.
No, eigenfunctions are used in a wide range of mathematical applications and can be introduced at various levels of mathematics education. They are not limited to advanced mathematics only.
Eigenfunctions can be applied in research by using them to model and analyze physical systems, such as quantum mechanics, fluid dynamics, and signal processing. They can also be used to solve differential equations and represent data in a more efficient manner.