MHB Finding Solutions for $y'=Ay$ with Vector and Matrix Components

[ineedhelpnow]

I think this should be my last question :D It's a question i missed on my last exam and have no idea how to do it and I wanted to quickly go over because my final is in a few hours so if anyone could help that would be awesome

View image: 20151215 092327

 

[evinda]

Hi! (Wave)

The coefficient matrix for this system is $A=\begin{bmatrix}
4 & 1\\
4 & 4
\end{bmatrix}$, for which we determine the eigenvalues $\lambda_1=6$ and $\lambda_2=2$, which are both positive real values.
What can we deduce from that?

 

[ineedhelpnow]

I'm thinking proper node that's unstable (Tmi)

 

[evinda]

I'm thinking proper node that's unstable (Tmi)

(Nod)

 

[ineedhelpnow]

How do i find the general solution of it though?

 

[evinda]

How do i find the general solution of it though?

Which is the form of the solution of $y'=Ay$ ?

 

[ineedhelpnow]

Right. Is the solution for both supposed to be found and then put together?

 

[evinda]

Right. Is the solution for both supposed to be found and then put together?

In this case, if we have $y'=Ay$, $y$ is a vector and $A$ is a matrix.
You can find from this the general solution.
Thus you will find to what the vector $y$ will be equal and so you will have also the solution of $y_1$ and $y_2$.