Final exam solutions, part II.

Definitions

#3

Qualitative assessment :  (a) and (c) have the form (2   0) + (0 & ... alue has eigenspace with negative slope, so stretching occurs roughly along the anti - diagonal .

a

Eigensystem[{{2, 1}, {-1, 2}}]

{{2 + , 2 - }, {{-, 1}, {, 1}}}

Show[kittym[MatrixExp[.3 {{2, 1}, {-1, 2}}]], DisplayFunction$DisplayFunction]

[Graphics:HTMLFiles/kittyfinal_8.gif]

⁃Graphics⁃

b

Eigensystem[{{2, -1}, {1, -2}}]

{{-3^(1/2), 3^(1/2)}, {{2 - 3^(1/2), 1}, {2 + 3^(1/2), 1}}}

Show[kittym[MatrixExp[.3 {{2, -1}, {1, -2}}]], DisplayFunction$DisplayFunction]

[Graphics:HTMLFiles/kittyfinal_13.gif]

⁃Graphics⁃

c

Eigensystem[{{2, -1}, {1, 2}}]

{{2 + , 2 - }, {{, 1}, {-, 1}}}

Show[kittym[MatrixExp[.3 {{2, -1}, {1, 2}}]], DisplayFunction$DisplayFunction]

[Graphics:HTMLFiles/kittyfinal_18.gif]

⁃Graphics⁃

d

Eigensystem[{{2, 1}, {-2, -1}}]

{{1, 0}, {{-1, 1}, {-1, 2}}}

Show[kittym[MatrixExp[.3 {{2, 1}, {-2, -1}}]], DisplayFunction$DisplayFunction]

[Graphics:HTMLFiles/kittyfinal_23.gif]

⁃Graphics⁃

Nonstarter

Eigensystem[{{-1, 2}, {2, -1}}]

{{-3, 1}, {{-1, 1}, {1, 1}}}

Show[kittym[MatrixExp[.3 {{-1, 2}, {2, -1}}]], DisplayFunction$DisplayFunction]

[Graphics:HTMLFiles/kittyfinal_28.gif]

⁃Graphics⁃

#4

c  (ii)

circnull = ParametricPlot[Evaluate[{{Cos[t], Sin[t]}, {t, t}}], {t, 0, π/2}, PlotRange {{0, 2}, {0, 2}}, AspectRatio1]

NDSolve[{x '[t] == 1 - x[t]^2 - y[t]^2, y '[t] x[t] - y[t], x[0] 2, y[0] 0}, {x[t], y[t]}, {t, 0, 10}]

RowBox[{{, RowBox[{{, RowBox[{RowBox[{x[t], , RowBox[{TagBox[RowBox[{InterpolatingFunc ... ox[{0., ,, 10.}], }}], }}], ,, <>}], ]}], False, Editable -> False], [, t, ]}]}]}], }}], }}]

ParametricPlot[{x[t], y[t]}/.%, {t, 0, 10}, PlotRange {{0, 2}, {0, 2}}, AspectRatio1]

Show[{%, circnull}]

[Graphics:HTMLFiles/kittyfinal_41.gif]

⁃Graphics⁃

NDSolve[{x '[t] ==    (x[t]^2 + y[t]^2) - 10, y '[t] x[t] - y[t], x[0] 2, y[0] 0}, {x[t], y[t]}, {t, 0, 5}]

RowBox[{{, RowBox[{{, RowBox[{RowBox[{x[t], , RowBox[{TagBox[RowBox[{InterpolatingFunc ... Box[{0., ,, 5.}], }}], }}], ,, <>}], ]}], False, Editable -> False], [, t, ]}]}]}], }}], }}]

ParametricPlot[{x[t], y[t]}/.%, {t, 0, 5}, PlotRange {{0, 2}, {0, 2}}, AspectRatio1]

Show[{%, circnull}]

[Graphics:HTMLFiles/kittyfinal_50.gif]

⁃Graphics⁃

#5

a

A is upper triangular, so the eigenvalues are -7, -6. The smaller eigenvalue,  -7, has eigenspace spanned by (1,0), so this system matches (vi).

b

A is lower triangular, so the eigenvalues are -9, -8. The larger eigenvalue,  -8, has eigenspace spanned by (0,1), so this system matches (iii).

c

dx/dt= 0 when y = (2 - 4x)/15 ;dy/dt= 0 when x = 2 or -2. Hence the equilibria are (2,- 2/5) and  (-2, 2/3) . The linearization at (2,- 2/5)  is   (-4      - 15 ... -4             0 , while the linearization at  
(-2, 2/3)   is (-4      - 15 ) 4             0

Eigensystem[{{-4, -15}, {-4, 0}}]

{{-10, 6}, {{5, 2}, {-3, 2}}}

Hence (2, -2/5) is a saddle, with the eigenspace of - 10 having positive slope and the eigenspace of 6 having negative slope .

Eigensystem[{{-4, -15}, {4, 0}}]

{{-2 + 2  14^(1/2), -2 - 2  14^(1/2)}, {{1/4 (-2 + 2  14^(1/2)), 1}, {1/4 (-2 - 2  14^(1/2)), 1}}}

Hence (-2, 2/3) is a spiral sink . The (downward sloping) eigenspace of the positive eigenvalue of the saddle should feed into the spiral sink, so this system matches (v) .

d

dx/dt= 0 when y = 1 or -1;dy/dt= 0 when y = -x/2. Hence the equilibria are (-2, 1) and (2, -1) . The linearization at (-2,1)  is   (0      - 2 ) 1         2 , while the linearization at  
(2, 1)   is (0       2) 1       2

Eigensystem[{{0, 2}, {1, 2}}]

{{1 + 3^(1/2), 1 - 3^(1/2)}, {{-1 + 3^(1/2), 1}, {-1 - 3^(1/2), 1}}}

Hence (2, 1) is a saddle, with the eigenspace of the positive eigenvalue  having positive slope and the eigenspace of the negative eigenvalue having negative slope .

Eigensystem[{{0, -2}, {1, 2}}]

{{1 + , 1 - }, {{-1 + , 1}, {-1 - , 1}}}

Hence (-2, 1) is a spiral source . The (downward sloping) eigenspace of the negative eigenvalue of the saddle should be fed by the spiral sink, so this system matches (i) .

Nonstarters

The other phase portraits have the wrong number of equilibria.

Created by Mathematica  (December 9, 2003)