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.