# Is the book incorrect here? Systems of diff eqs.

1. Nov 19, 2012

### Zondrina

1. The problem statement, all variables and given/known data

Here is the question from my book along with their solution to the problem :

Part 1 : http://gyazo.com/6467751155edcefec6cc583d164d2ae7
Part 2 : http://gyazo.com/d91752f0a18a72af96f6afc74117f011
Part 3 : http://gyazo.com/7a17dbca3e414d0ddc8458abce2eda9c

2. Relevant equations

I'll denote the coefficient matrix by the letter A.
I denote my null space operator by N.
t is the transpose of a vector.

3. The attempt at a solution

Now I'm almost positive the book is wrong here. Here's my solution.

Assume that $x = εe^{λt}$ so our equation becomes :

$λεe^{λt} = Aεe^{λt} \Rightarrow (A - λI)ε = 0$

So we seek eigenvector(s) ε such that the above equation is satisfied. So lets find the eigenvalues for A ( I'll skip some of the boring algebra here ) :

$det(A - λI) = (λ-3)(λ+1)$ Thus λ1 = 3 and λ2 = -1 are eigenvalues for A.

Now to find the eigenvectors for the eigenvalues, we seek the null space of A - λjI. So :

$ε(λ_1) = N(A - 3I) = span\left\{{[1/2 \space \space 1]^t}\right\}$

$ε(λ_2) = N(A + I) = span\left\{{[-1/2 \space \space 1]^t}\right\}$

See ^ for some reason I get different eigenvectors than the book does. Everything else that follows after this is exactly the same as the book except my eigenvectors.

Have I made an error somewhere? Or is the book wrong in this case? Some clarification on this would be greatly appreciated.

Thanks in advance.

Last edited: Nov 19, 2012
2. Nov 19, 2012

### Ray Vickson

No, you are getting the same eigenvectors, but just scaled differently. Instead of <1/2,1> the book uses <1,2>, and instead of <-1/2,1> the book uses <1,-2>. Eigenvectors are, of course, not unique: any nonzero scalar multiple of an eigenvector is also an eigevector for the same eigenvalue.

RGV

3. Nov 19, 2012

### Zondrina

Ahh yes, I remember now. So multiply my first one by 2 and my second one by -2 to get the same answer as they have.

Thanks for clearing that up for me RGV.

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