Question

Multiple Choice: Let A = $\begin{bmatrix} -1 & 0 & -1\\ 0 & 1 & 0\\ 4 & 0 & -1 \end{bmatrix}$ . Let x be the solution of the following initial value problem:

x' = Ax, x(0) = $\begin{bmatrix} 1\\ 1\\ 1\end{bmatrix}$ .

What is the value of ln(x($\pi$))?

(a) $\begin{bmatrix} -\pi \\ -\pi \\ \pi \end{bmatrix}$    (b) $\begin{bmatrix} 2\pi \\ 0 \\ -\pi \end{bmatrix}$    (c) $\begin{bmatrix} -\pi \\ 0 \\ 1 \end{bmatrix}$    (d) $\begin{bmatrix} \pi \\ -\pi \\ 3\pi \end{bmatrix}$    (e) $\begin{bmatrix} -\pi \\ \pi \\ -\pi \end{bmatrix}$

Answer 1

The answer is option(d). Look at the handwritten detailed solution.

Few Points:

1. If Roots of complimentary equation is of the form (a $\pm$ i*b) (i.e complex), then the solution is of the form

   x(t) = $e^{-at}(Fsin(bt)+Dcos(bt))$ . This is used in finding the solution.

We have A-[-o-17 I 4 0 - 1 differential We have the equation. X Co x = AX . I withe initial value Let then x's dat 92 I dal 93 here, a, a, diz of t (lets Wrilling are function say.) out the I at J explicitly matrices we have da, 1 no 1 1 CIL da dt 13 야 - O i di HILL - d dil at daz dt 하 a = -80 -83-07 where à, , ,aj, a; ane å = 22-0 de dar and data respectively. dt at a = 40, -23-0l Taking eq (2) cup have, ► de dt Now when t= 0 a 2 = 1

Now Cup have two coupled differential equation aj: -4,-23 – ☺ az = 48,-43.- Now, Jorz = -a, -i, -o n Putting in we have (-2, - ) - 49,- (-2,-6, ) da dt TTTTTTTTT ITTEINTE we have D + 20+5 -0 ID=-21 4-20 - - tai form This is the When roots are complex 2 *D = -2tui » 10 = =+21 Hence, a, (t) = et (Asinat + B cosat) & a. (o) = thus 2,16) = f(1519 24 4 605 24) & 1B = 1 From ② we have az = -a, - s) az é +(986024 + cos2+) - [-et (Asin 24° + Cos 2+) tput (2A Cosat Basinat) -a3 = -26** (A cos ge- Sinat)? Noc, az (0) = 1 - 2A 1 - 1A = = 2 2 1 10. We have put We have put value of A in 2, (+) and az (4)

Thus, x, (t) = pt (cosat - Sinat , as (t) - ae* (cosat +Sigat) I general Solution NOU *, (*)=en az (a) = en az (a) = 26 x 7 = et en (alt)) = (en Ca (A)) Thus en (*2 (x) ) en (az (a) ) » en Caca)) fen (en) 7 len(er) o fa com)