Help with a Problem Involving Eigenvalues and Exponential Functions

[PBJinx]

1.[tex]\frac{dx}{dt}[/tex]= [tex]\stackrel{9 -12}{2 -1}[/tex]

x(0)=[tex]\stackrel{-13}{-5}[/tex]

So I seem to be having issues with this problem

There are 2 eigenvalues that I obtained from setting

Det[A-rI]=0

That gave me [tex]r^{2}-8r+15=0[/tex]

solving for r and finding the roots i got

(r-3)*(r-5)=0

so the roots are [tex]r_{1}=3[/tex] and [tex]r_{2}=5[/tex]

putting those back into [A-rI] i obtained

[tex]r_{1}[/tex]

4y-12z=0
2y-6z=0

so the vector [tex]w_{1}=\stackrel{2}{1}[/tex]

for [tex]r_{2}[/tex] i obtained

4y-12z=0
2y-6z=0

so [tex]w_{2}=\stackrel{3}{1}[/tex]

I am now left with this equation

v(t)=[W][[tex]e^{t\Lambda}[/tex]c

Where c=[[tex]W^{-1}[/tex][tex]v_{0}[/tex]

that leads to finding [tex]W^{-1}[/tex] where W=[tex]\stackrel{2 3}{1 1}[/tex]

[tex]W^{-1}=\stackrel{-1 3}{1 -2}[/tex]

c=[[tex]\stackrel{2}{3}[/tex]

I put back into my equation and get

V(t)=[tex]\stackrel{4e^{3t} + 6e^{5t}}{2e^{3t}+3e^{5t}}[/tex]


i put that into webwork and i get an incorrect answer

any help?

 

[lanedance]

do you mean
[tex]\begin{pmatrix} x_1' \\ x_2' \end{pmatrix} = \begin{pmatrix} 9 & -12 \\ 1 & -2\end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} [/tex]

 

[PBJinx]

do you mean
[tex]\begin{pmatrix} x_1' \\ x_2' \end{pmatrix} = \begin{pmatrix} 9 & -12 \\ 1 & -2\end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} [/tex]

yes, sorry i am not used to using typing it out this way

 

[lanedance]

just a little hard to work out exactly what youre doing, I'm not too sure what you did with the wronksian

isn't the general solution
[tex] \textbf{v(t)} = \textbf{w}_1 e^{3t}+ \textbf{w}_2 e^{5t}[/tex]

 

[PBJinx]

just a little hard to work out exactly what youre doing, I'm not too sure what you did with the wronksian

isn't the general solution
[tex] \textbf{v(t)} = \textbf{w}_1 e^{3t}+ \textbf{w}_2 e^{5t}[/tex]

thank you for the help. i went to the professor today and figured it out with him.