Help with a Problem Involving Eigenvalues and Exponential Functions

[PBJinx]

1.\frac{dx}{dt}= \stackrel{9 -12}{2 -1}

x(0)=\stackrel{-13}{-5}

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 r^{2}-8r+15=0

solving for r and finding the roots i got

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

so the roots are r_{1}=3 and r_{2}=5

putting those back into [A-rI] i obtained

r_{1}

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

so the vector w_{1}=\stackrel{2}{1}

for r_{2} i obtained

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

so w_{2}=\stackrel{3}{1}

I am now left with this equation

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

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

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

W^{-1}=\stackrel{-1 3}{1 -2}

c=[\stackrel{2}{3}

I put back into my equation and get

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


i put that into webwork and i get an incorrect answer

any help?

 

[lanedance]

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

 

[PBJinx]

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

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
\textbf{v(t)} = \textbf{w}_1 e^{3t}+ \textbf{w}_2 e^{5t}

 

[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
\textbf{v(t)} = \textbf{w}_1 e^{3t}+ \textbf{w}_2 e^{5t}

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