Solve the following problems:
1. .
If x=dx/dt
a. Explain the relationship between x· and x relative to x increasing/decreasing.
b. Explain the relationship between x·· and x· relative to x· increasing/decreasing.
c. Explain the relationship between x·· and concavity, relative to x vs t.
d. Define an inflection point, how it is found and its relevance to x vs t graph and show its relevance to x·.
2. Solve by separation of variables and plot the solution of the ODE:
dx
— x· = a1x–2; t≥1,x(1)=3/α1,α1>2,x≥1.
dt
3. Solve the equation:x·=-Rln(x),t≥0,x>0,R>0,x(0)=x0·
What is the asymptotic value of x as t→ ∞ Explain
4. For the given graph, re-sketch a graph for x01 and indicate the system progress (using arrows and explain) from x01( as t ->∞.
a. Identify any critical points as stable, unstable, or semi-stable—explain.
b. Repeat for x02, x03, and x04.
5. Draw and label the phase diagram xvsx.
a. Indicate the system progress using arrows and explain.
b. Identify any critical points as stable, unstable, or semi-stable and explain.
x· = —α1x + α2, α1> 0, α2> O.
Draw a family of solution curves xvst.
6. Draw and label the phase diagram x·vs x for x· = 3x2 —9x.
a. Indicate the system progress using arrows and explain.
b. Identify any critical points as stable, unstable, or semi-stable and explain.
c. Find the x value, xm, where x· is minimum or maximum.
d. Plot x vs t for x(0) = xo> xm, . There should be 2different cases for xo > Xm.
7. A fish farm is allowed to reach a stable level before any fishing is allowed. R, in tons per year, is the stocking rate for the farm. The growth rate is modeled by x· = α1x-α2x2+R;a1 > 0, a2 > 0, R> 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 and explain.
c. Draw the phase diagram for R = 0.
d. On one coordinate system plot, draw and label the phase diagrams for different values of R: R=0, R=Rmax. and 0123max.
e. What are the units α1 αl and α2?