Separation of Variables, System Progress, and Phase Diagrams
Solve the following problems:
1. Solve by separation of variables and plot the solution to the ODE: dx/dt =x·
x· =4x-2;t≥1,x(1)=3/4
2. Solve the equation x·= —x In (x), t ≥ 0, x(0)= x0.
What is the asymtotic value of x at t-> ∞. EXPLAIN your answer.
3. For the given graph, re-sketch a graph for x01 and indicate the system progress (using arrows and EXPLAIN) from x01 as t-> ∞op. Identify any critical points as stable, unstable, or semi-stable (EXPLAIN). Repeat for X02 x03, and X04.
![1223_phase diagram.jpg](https://secure.tutorsglobe.com/CMSImages/1223_phase diagram.jpg)
4. Draw and label the phase diagram (x· vs. x). Indicate the system progress (using arrows and EXPLAIN). Identify any critical points as stable, unstable, or semi-stable (EXPLAIN): x· = a1x - a2, a1, >0, a2 >0.
5. Draw and label the phase diagram ((x· vs. x) for x· = 3x2 - 9x. Indicate the system progress (using arrows and EXPLAIN). Identity any critical points as stable, unstable, or semi-stable (EXPLAIN).
6. A harvesting model is represented by: x· = a1x - a2x2 - h; a1 > 0, a2 > 0, h> 0.
a. Draw the phase diagram ( (x· vs.x) and label the values of the intercepts and the point of maximum x·
b. Identify any critical points as stable, unstable, or semi-stable (EXPLAIN).
c. What values of h doom the crop to complete depletion (EXPLAIN)?
d. What is the maximum value of h that produces a critical crop level (EXPLAIN).
e. Plot x· vs. x for maximum h and determine if the critical point is stable, unstable, or semi-stable (EXPLAIN).
f. How does the maximum value of h change if a1, the reproduction rate parameter, increases (EXPLAIN)?
g. How does the maximum value of h change if a1, the reproduction rate parameter, decreases (EXPLAIN)?
7. x· = a1,x - a2 If x has units of ounces and x· has units of ounces per second, what are the units of a1 and a2 (EXPLAIN)?