Thread: 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!

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?

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?

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 for every eigenspace. Now, if is the matrix whose columns are the eigenvalues associated to respectively, the solution is