# Math 53 Groupwork

Note: groupwork will not be graded. It should be "fun".

### Groupwork #1, Jan. 8, 1999

(Note: postscript files can be viewed with ghostview and printed with lpr.)

### Groupwork #2, 1/12.

Edwards and Penney, section 1.5: problems 6 and 22; section 1.6: problem 3.

### Groupwork #3, 1/14.

The population of fish in a lake (from which fish are harvested) is described by the equation

dx/dt = kx(M-x)-h.

Here M is the carrying capacity of the lake, kM is the natural growth rate, and h is the rate at which fish are being harvested.

(a) What is the largest sustainable yield h? (I.e. what is the largest h such that there are solutions to the above equation with x>0 for all t>0.) Call this h_max. This number depends on k and M.

(b) Give a rough sketch of the solution curves for h less than h_max, h=h_max, and h greater than h_max.

(c) If this is an approximate model of a real-world lake, is it a good idea to fish near the maximum sustainable rate h_max? Explain.

### Groupwork #4, 1/21.

Solve y''-3y'+2y=x, y(0)=3/4, y'(0)=-1/2.

### Groupwork #5, 1/28.

Solve y''+2y=(cos x)^2, y(0)=1, y'(0)=2 sqrt{2}.

### Groupwork #6, 2/3.

E&P section 4.2, number 6.

### Groupwork #7, 2/16.

E&P section 5.2, number 5. Solve with the initial condition (11,-1).

### Groupwork #8, 2/23.

For the nonlinear systems in E&P section 6.1 problems 7,8: find the critical points. Determine as much as you can about their type and stability by linearizing the equations. Try to draw a picture of the trajectories (without looking at the answers in the book.)

### Groupwork #9, 3/3.

E&P 7.2 # 4, 7.4 # 33.