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Problem Set 9

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Problem Set 9

  1. Maximize $ f(x,y)=x^2 + y^2$ subject to the constraints $ 2x+y \leq2$ , $ x\geq 0$ , $ y\geq 0$ .

  2. Maximize $ h\left( x,y\right) =2x^{2}-y^{2}$ subject to
    $\displaystyle x^{2}+y^{2}$ $\displaystyle =$ $\displaystyle 1$  
    $\displaystyle x,y$ $\displaystyle \geq$ 0  

  3. Use a phase diagram to analyze:
    1. $ \overset{\cdot }{y}=-y+y^{2}+\frac{3}{16}$
    2. $ \overset{\cdot }{y}=y-y^{\frac{1}{2}}$

  4. For the matrix

    $\displaystyle A=\left[ \begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array} \right]$    

    1. Write the characteristic equation and find the characteristic roots
    2. Find the eigenvectors corresponding to the characteristic equation.
    3. Repeat for the matrix

      $\displaystyle A=\left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \end{array} \right]$    

  5. Draw the phase diagram and conduct stability analysis for the following:

    1. $\displaystyle \overset{\cdot }{y_{1}}$ $\displaystyle =$ $\displaystyle 2y_{1}-9y_{2}$  
      $\displaystyle \overset{\cdot }{y_{2}}$ $\displaystyle =$ $\displaystyle -3y_{1}-4y_{2}$  


    2. $\displaystyle \overset{\cdot }{y_{1}}$ $\displaystyle =$ $\displaystyle 2y_{1}-9y_{2}+35$  
      $\displaystyle \overset{\cdot }{y_{2}}$ $\displaystyle =$ $\displaystyle -3y_{1}-4y_{2}+70$  

  6. Consider the problem discussed in class:

    $\displaystyle \overset{\cdot}{x} = a_{11}x + a_{12}y + c_{1}$    
    $\displaystyle \overset{\cdot}{y} = a_{21}x + a_{22}y + c_{2}$    

    1. Consider the case where $ \overset{\cdot}{x}$ is flatter than $ \overset{\cdot}{y}$ . Under what conditions will this be an unstable focus? An unstable node?
    2. Consider the case where $ \overset{\cdot}{y}$ is flatter than $ \overset{\cdot}{x}$ . What can you conclude about the steady state?


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Jennifer Anne Thacher 2008-12-04
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