Solving an Initial Value Problem with a Matrix and a Given Vector: A Case Study

[Shackleford]

x' = Ax, x(0) = v,

where A is the matrix in problem 6 and v = [1,2]^T. Do not use the eigenvalues and the eigenvectors of A. (Hint A^2v = 0).A = [ (1, -1)^T (1, -1)^T ]

All I did was calculate e^(tA)v = v[I + tA].

In this case, an IVP, x(t) is the solution. x(t) = e^(tA)v.

Since I wasn't asked to find a fundamental set of solutions, I didn't need to calculate e^(tA)v for every v that's a basis in R^n, right?

 

[Mark44]

Your solution looks good to me, and working with the exponential was the way to go. That's the reason for hint, I'm pretty sure.

e^(tA) = I + tA =
[t+1 t]
[-t -t+1]
solves the differential equation, and x(0) = v, so all is good.

 

[Shackleford]

Your solution looks good to me, and working with the exponential was the way to go. That's the reason for hint, I'm pretty sure.

e^(tA) = I + tA =
[t+1 t]
[-t -t+1]
solves the differential equation, and x(0) = v, so all is good.

Yeah. I figured that. I guess I'm a little fuzzy on why you don't have to use the general procedure. They have a little proof in the book which is fairly straightforward. I understand that.

 

[Shackleford]

Well, I got my exam back today after the 2.5-hour long final.

I only got 5/15 for that work. Apparently, and rightfully so, I had the v on the wrong side. I Also forgot to to put the v in tAv.

I want to email the professor for points on that problem and another one. Are you sure I did it correctly? I'll probably scan the questions and my work to see if I should ask for more points.