Draw and label the phase diagramx vs x for x 3x2 - 9x

Question

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 ln(x),t≥0, x(0) = xo

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→∞. Identify any critical points as stable, unstable, or semi-stable (EXPLAIN). Repeat for x₀₂ , x₀₃, and x₀₄.

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). Identify 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 ( (ivs.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 al, the reproduction rate parameter, increases (EXPLAIN)?

g. How does the maximum value of h change if al, the reproduction rate parameter, decreases (EXPLAIN)?

7. x^. = a₁x-a₂ If x has units of ounces and x^. has units of ounces per second, what are the units of a1 and a2 (EXPLAIN)?