Solve d2x dt2 = Ax, x(0) = x0, dx dt (0) = x_ 0. a. A = _ 1 5 2 4 _ , x0 = _ 7 0 _ , x_

QUESTION:

Solve \(\frac{d^2x}{dt^2}=Ax,\ x(0)\ =\ x_0,\ \frac{dx}{dt}(0)\ =\ x_0'\).

a. \(A=\left[\begin{array}{ll} 1 & 5 \\ 2 & 4 \end{array}\right],\ \mathbf{x}_{0}=\left[\begin{array}{l} 7 \\ 0 \end{array}\right],\ \mathbf{x}_{0}^{\prime}=\left[\begin{array}{r} -5 \\ 2 \end{array}\right]\)

b. \(A=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{r} -2 \\ 2 \end{array}\right], \mathbf{x}_{0}^{\prime}=\left[\begin{array}{l} 1 \\ 3 \end{array}\right]\)

c. \(A=\left[\begin{array}{ll} 1 & 3 \\ 3 & 1 \end{array}\right],\ \mathbf{x}_{0}=\left[\begin{array}{r} -2 \\ 4 \end{array}\right],\ \mathbf{x}_{0}^{\prime}=\left[\begin{array}{l} 2-3 \sqrt{2} \\ 2+3 \sqrt{2} \end{array}\right]\)

d. \(A=\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right], \mathbf{x}_{0}^{\prime}=\left[\begin{array}{l} 2 \\ 1 \end{array}\right]\)