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Mathematics LibreTexts

Section 10.2 Answers

  • Page ID
    30065
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    1. 

    1. \(\bf{y'}=\left[\begin{array}{cc}{2}&{4}\\{4}&{2}\end{array}\right]{\bf y}\)
    2. \(\bf{y'}=\left[\begin{array}{cc}{-2}&{-2}\\{-5}&{1}\end{array}\right]{\bf y}\)
    3. \(\bf{y'}=\left[\begin{array}{cc}{-4}&{-10}\\{3}&{7}\end{array}\right]{\bf y}\)
    4. \(\bf{y'}=\left[\begin{array}{cc}{2}&{1}\\{1}&{2}\end{array}\right]{\bf y}\)

    2. 

    1. \(\bf{y'}=\left[\begin{array}{ccc}{-1}&{2}&{3}\\{0}&{1}&{6}\\{0}&{0}&{-2}\end{array}\right]{\bf y}\)
    2. \(\bf{y'}=\left[\begin{array}{ccc}{0}&{2}&{2}\\{2}&{0}&{2}\\{2}&{2}&{0}\end{array}\right]{\bf y}\)
    3. \(\bf{y'}=\left[\begin{array}{ccc}{-1}&{2}&{2}\\{2}&{-1}&{2}\\{2}&{2}&{-1}\end{array}\right]{\bf y}\)
    4. \(\bf{y'}=\left[\begin{array}{ccc}{3}&{-1}&{-1}\\{-2}&{3}&{2}\\{4}&{-1}&{-2}\end{array}\right]{\bf y}\)

    3. 

    1. \(\bf{y'}=\left[\begin{array}{cc}{1}&{1}\\{-2}&{4}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{1}\\{0}\end{array}\right]\)
    2. \(\bf{y'}=\left[\begin{array}{cc}{5}&{3}\\{-1}&{1}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{9}\\{-5}\end{array}\right]\)

    4. 

    1. \(\bf{y'}=\left[\begin{array}{ccc}{6}&{4}&{4}\\{-7}&{-2}&{-1}\\{7}&{4}&{3}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{3}\\{-6}\\{4}\end{array}\right]\)
    2. \(\bf{y'}=\left[\begin{array}{ccc}{8}&{7}&{7}\\{-5}&{-6}&{-9}\\{5}&{7}&{10}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{2}\\{-4}\\{3}\end{array}\right]\)

    5. 

    1. \({\bf y'}=\left[\begin{array}{cc}{-3}&{2}\\{-5}&{3}\end{array}\right] + \left[\begin{array}{c}{3-2t}\\{6-3t}\end{array}\right]\)
    2. \({\bf y'}=\left[\begin{array}{cc}{3}&{1}\\{-1}&{1}\end{array}\right]{\bf y} + \left[\begin{array}{c}{-5e^{t}}\\{e^{t}}\end{array}\right]\)

    10. 

    1. \(\frac{d}{dt}Y^{2}=Y'Y+YY'\)
    2. \(\frac{d}{dt}Y^{n}=Y'Y^{n-1}+YY'Y^{n-2}+Y^{2}Y'Y^{n-3}+\ldots + Y^{n-1}Y'=\sum_{r=0}^{n-1}Y^{r}Y'Y^{n-r-1}\)

    13. \(B=(P'+PA)P^{-1}\)