Consider the equation
1. Find the trace ? and the determinant ?. Sketch the graphs of ? vs. b and ? vs. b. Then sketch the graph of the curve ? = ? (b), ? = ?(b) in the ??-plane along with ?^2 = 4?, ? = 0 and ? = 0(? ? 0).
2. Find the intersection points of the the curve ? = ?(b),? = ?(b) with ?^2 = 4?, ? = 0 and ? = 0(? ? 0) and the corresponding values of b. The corresponding values of b are called the critical values.
3. Find the general solution for each critical value.
4. Draw a phase portrait for a value slightly below the smallest critical value and a value slightly above the largest critical value. (You can pick any value slightly below the smallest critical value and any value slightly above the largest critical value)
1) the trace is (5/4 + b) + 5/4 = 5/2 + b. Determinant is (5/4+b)*5/4 - 5/4*b = 25/16.
The sketches:
(i)
(ii) Determinant value
(2) The intersection of T2 = 4 => (5/4+b)2 = 4*25/16 = 25/4
=> 25/16 + 5/2b + b2 = 25/4
=> 16b2 + 40b - 75 = 0
the roots are b = ; b = 5/4 , 15/4
The graph is,
(3) For any general T2 = k*, we have,
(5/4 + b)2 = k*25/16
=> 16b2 + 40b + 25(1-k) = 0
the general solution will be, b = .
Consider the equation 1. Find the trace ? and the determinant ?. Sketch the graphs of...
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