14.49: Section 10.3 Answers

2. \({\bf y'}=\left[\begin{array}{cc}{0}&{1}\\[4pt]{-\frac{P_{2}(x)}{P_{0}(x)}}&{-\frac{P_{1}(x)}{P_{0}(x)}}\end{array}\right]{\bf y}\)

3. \({\bf y'}=\left[\begin{array}{cccc}{0}&{1}&{\ldots }&{0}\\[4pt]{\vdots}&{\vdots}&{\ddots}&{\vdots}\\[4pt]{0}&{0}&{\ldots}&{1}\\[4pt]{-\frac{P_{n}(x)}{P_{0}(x)}}&{-\frac{P_{n-1}(x)}{P_{0}(x)}}&{\ldots}&{-\frac{P_{1}(x)}{P_{0}(x)}}\end{array}\right]{\bf y}\)

7.

b. \({\bf y}=\left[\begin{array}{c}{3e^{6t}-6e^{-2t}}\\[4pt]{3e^{6t}+6e^{-2t}}\end{array}\right]\)

c. \({\bf y}=\frac{1}{2}\left[\begin{array}{cc}{e^{6t}+e^{-2t}}&{e^{6t}-e^{-2t}}\\[4pt]{e^{6t}-e^{-2t}}&{e^{6t}+e^{-2t}}\end{array}\right]{\bf k}\)

8.

b. \({\bf y}=\left[\begin{array}{c}{6e^{-4t}+4e^{3t}}\\[4pt]{6e^{-4t}-10e^{3t}}\end{array}\right]\)

c. \({\bf y}=\frac{1}{7}\left[\begin{array}{cc}{5e^{-4t}+2e^{3t}}&{2e^{-4t}-2e^{3t}}\\[4pt]{5e^{-4t}-5e^{3t}}&{2e^{-4t}+5e^{3t}}\end{array}\right]{\bf k}\)

9.

b. \({\bf y}=\left[\begin{array}{c}{-15e^{2t}-4e^{t}}\\[4pt]{9e^{2t}+2e^{t}}\end{array}\right]\)

c. \({\bf y}=\left[\begin{array}{cc}{-5e^{2t}+6e^{t}}&{-10e^{2t}+10e^{t}}\\[4pt]{3e^{2t}-3e^{t}}&{6e^{2t}-5e^{t}}\end{array}\right]{\bf k}\)

10.

b. \({\bf y}=\left[\begin{array}{c}{5e^{3t}-3e^{t}}\\[4pt]{5e^{3t}+3e^{t}}\end{array}\right]\)

c. \({\bf y}=\frac{1}{2}\left[\begin{array}{cc}{e^{3t}+e^{t}}&{e^{3t}-e^{t}}\\[4pt]{e^{3t}-e^{t}}&{e^{3t}+e^{t}}\end{array}\right]{\bf k}\)

11.

b. \({\bf y}=\left[\begin{array}{c}{e^{2t}-2e^{3t}+3e^{-t}}\\[4pt]{2e^{3t}-9e^{-t}}\\[4pt]{e^{2t}-2e^{3t}+21e^{-t}}\end{array}\right]\)

c. \({\bf y}=\frac{1}{6}\left[\begin{array}{ccc}{4e^{2t}+3e^{3t}-e^{-t}}&{6e^{2t}-6e^{3t}}&{2e^{2t}-3e^{3t}+e^{-t}}\\[4pt]{-3e^{3t}+3e^{-t}}&{6e^{3t}}&{3e^{3t}-3e^{-t}}\\[4pt]{4e^{2t}+3e^{3t}-7e^{-t}}&{6e^{2t}-6e^{3t}}&{2e^{2t}-3e^{3t}+7e^{-t}}\end{array}\right]{\bf k}\)

12.

b. \({\bf y}=\frac{1}{3}\left[\begin{array}{c}{-e^{-2t}+e^{4t}}\\[4pt]{-10e^{-2t}+e^{4t}}\\[4pt]{11e^{-2t}+e^{4t}}\end{array}\right]\)

c. \({\bf y}=\frac{1}{3}\left[\begin{array}{ccc}{2e^{-2t}+e^{4t}}&{-e^{-2t}+e^{4t}}&{-e^{-2t}+e^{4t}}\\[4pt]{-e^{-2t}+e^{4t}}&{2e^{-2t}+e^{4t}}&{-e^{-2t}+e^{4t}}\\[4pt]{-e^{-2t}+e^{4t}}&{-e^{-2t}+e^{4t}}&{2e^{-2t}+e^{4t}}\end{array}\right]{\bf k}\)

13.

b. \({\bf y}=\left[\begin{array}{c}{3e^{t}+3e^{-t}-e^{-2t}}\\[4pt]{3e^{t}+2e^{-2t}}\\[4pt]{-e^{-2t}}\end{array}\right]\)

c. \({\bf y}=\left[\begin{array}{ccc}{e^{-t}}&{e^{t}-e^{-t}}&{2e^{t}-3e^{-t}+e^{-2t}}\\[4pt]{0}&{e^{t}}&{2e^{t}-2e^{-2t}}\\[4pt]{0}&{0}&{e^{-2}}\end{array}\right]{\bf k}\)

14. \(YZ^{-1}\) and \(ZY^{-1}\)