Eigenvalue Method as a diferential equation!

[wolly]

How did m+nt appear and how did n=0 and the other things appear? Also who is [math]μ[/math]?
I'm talking about an Eigenvalue Method as a differential equation and I don't know why this is solved like this.Can someone help me?

 

[blamocur]

I don't understand the context, and I cannot read Romanian, but here is what I think about your first question:

[imath]m+nt[/imath] appears because [imath]\lambda_1[/imath], which is equal to -3, has multiplicity of 2.

 

[wolly]

I don't understand the context, and I cannot read Romanian, but here is what I think about your first question:

[imath]m+nt[/imath] appears because [imath]\lambda_1[/imath], which is equal to -3, has multiplicity of 2.

And n=0 c=-a-b r=d=s
IT says that IT results from Y'=AY?

 

[wolly]

I don't understand the context, and I cannot read Romanian, but here is what I think about your first question:

[imath]m+nt[/imath] appears because [imath]\lambda_1[/imath], which is equal to -3, has multiplicity of 2.

Do You have Something like a PDF that can show me how to solve this?

 

[wolly]

And n=0 c=-a-b r=d=s
IT says that IT results from Y'=AY?

@blamocur How did these 3 solutions come from?

@logistic_guy A Little help here!

 

[blamocur]

And n=0 c=-a-b r=d=s

Not sure about [imath]n[/imath], but the rest is derived from the fact that (a,b,c) and (d,r,s) are eigenvectors of A.

 

[blamocur]

Do You have Something like a PDF that can show me how to solve this?

No, I don't have it. But it looks like pretty standard stuff about ODEs with constant coefficients. Don't you have a textbook? Where does this problem come from? Is it part of your homework?

 

[wolly]

No, I don't have it. But it looks like pretty standard stuff about ODEs with constant coefficients. Don't you have a textbook? Where does this problem come from? Is it part of your homework?

IT comes from a math college contest exercise book and I know how to solve some parts but not all of them!
I am referring to the eigenvalue and eigenvector Matrix Equations

 

[blamocur]

I am referring to the eigenvalue and eigenvector Matrix Equations

You need to learn the subject in a more systematic way. Asking for help on a forum is no substitute for studying.

 

[wolly]

I found a pdf of how to solve these. I forgot to add them
@blamocur

 

[fresh_42]

The idea is as follows. If we have [imath] (A-\lambda I)(x)=0, [/imath] i.e. [imath] Ax=\lambda x [/imath] then we solve [imath] y'=\lambda y [/imath] and obtain [imath] y=me^{\lambda t}. [/imath] However, we actually have for [imath] \lambda=-3 [/imath] that
[math] (A-\lambda I)^2(x)=0. [/math] That means that either [imath] (A-\lambda I)(x)=0 [/imath] is already zero, in which case we get the solution [imath] y=me^{\lambda t}, [/imath] or, and this is different, that [imath] (A-\lambda I)((A-\lambda I)(x))=0.[/imath] Now
[math] (A-\lambda I)(A-\lambda I)=A^2-2\lambda A +\lambda^2 I=0 [/math]and we have to solve [imath] y''-2\lambda y'+\lambda^2y=0. [/imath] Let's see if [imath] y=nte^{\lambda t} [/imath] is a solution for this equation.

[math]\begin{array}{lll} y&=n te^{\lambda t}\\ y'&=ne^{\lambda t}+n\lambda te^{\lambda t}=n e^{\lambda t}+\lambda y\\ y''&=n\lambda e^{\lambda t}+\lambda y'=2n\lambda e^{\lambda t}+\lambda^2 y \end{array}[/math][math]\begin{array}{lll} y''-2\lambda y'+\lambda^2 y&=2n\lambda e^{\lambda t}+\lambda^2 y-2\lambda(n e^{\lambda t}+\lambda y)+\lambda^2 y=0 \end{array}[/math]
Hence, it is a solution. Combining both, we get
[math] y=me^{\lambda t}+n te^{\lambda t}=(m+nt)e^{\lambda t}.[/math]This is the solution space for the generalized eigenspace [imath] \lambda=\lambda_1=-3 [/imath] with multiplicity two.
If we finally add the solution space for the generalized eigenspace [imath] \lambda=\lambda_2=0 [/imath] with multiplicity one, then we get
[math] y=(m+nt)e^{\lambda_1 t}+pe^{\lambda_2 t}=(m+nt)e^{-3t}+pe^{0\cdot t}=(m+nt)e^{-3t}+p. [/math]The parameters [imath] m,n,p [/imath] result from the initial values for the differential equation.

 

[logistic_guy]

@blamocur How did these 3 solutions come from?

@logistic_guy A Little help here!

Forgive me @wollyI was not here. I was visiting the \(\displaystyle \text{USA}\). Read post #\(\displaystyle 11\), professor Stefan explained it perfectly.