Is the book incorrect here? Systems of diff eqs.

[STEMucator]

Homework Statement

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

Homework 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.

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 let's 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 - λ_jI. 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.

 

[Ray Vickson]

Homework Statement

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

Homework 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.

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 let's 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 - λ_jI. 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.

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

 

[STEMucator]

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

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.