Math Help - Initial-value problem using the method of eigenvalues/eigenvectors (Answer included).

1. Initial-value problem using the method of eigenvalues/eigenvectors (Answer included).

The biggest challenge for me in attempting this problem is using the Linear algebra.

If I'm correct, I must:

1) Find eigenvalues.
2) Find eigenvectors.
3) Compute e^(At)
4) Multiply e^(At) by y_0 = y(0)

So, assuming what I said is correct, it seems like I know what I am doing but it's in setting up each step and then taking the result and moving to the next step that loses me so I would really appreciate if someone could show me how to set up each step and the answer each step gives to feed into the next step. I don't need mechanical computations (software outputs should most likely be fine).

Any help would be GREATLY appreciated!
Thanks in advance!

2. Re: Initial-value problem using the method of eigenvalues/eigenvectors (Answer includ

Originally Posted by s3a
If I'm correct, I must:
1) Find eigenvalues.
2) Find eigenvectors.
3) Compute e^(At)
4) Multiply e^(At) by y_0 = y(0)
That is right, so let us go with 1) and 2). What eigenvalues and eigenvectors did you obtain?

3. Re: Initial-value problem using the method of eigenvalues/eigenvectors (Answer includ

Sorry for the late reply, I was having difficulty computing things. I'm attaching my work so far. Hopefully, it's correct.

Also, I know this is minor in comparison to my other problems but, instead of u_1, u_2, u_3 and w_1, w_2, w_3, should I have reused the v_i notation (strictly speaking) or was I technically correct to use different variables?

4. Re: Initial-value problem using the method of eigenvalues/eigenvectors (Answer includ

Originally Posted by s3a
Also, I know this is minor in comparison to my other problems but, instead of u_1, u_2, u_3 and w_1, w_2, w_3, should I have reused the v_i notation (strictly speaking) or was I technically correct to use different variables?
It is irrelevant in our case. You also can use (for example) the notation $x_1,x_2,x_3$ for every eigenspace. Now, if $P$ is the matrix whose columns are the eigenvalues associated to $4,-1,1$ respectively, the solution is $\vec{x}(t)=e^{tA}\vec{x}(0)=P\;\textrm{diag}\;(e^{ 4t},e^{-t},e^t)\;P^{-1}\vec{x}(0)=\ldots$