MATH2270

FW00

Assignment 1 Due: Wednesday, Mar. 14, 2001

1. (a) For each of the following differential equations find the analytic family of solutions which satisfies the differential equation. Also find the explicit solution which satisfies the initial conditions indicated:

(i) y = 5 when x = 2

(ii) y = 2 when x = 0

(iii) y = 1 when x = 0

(iv) y = 1 when x = 1

(b) For each of the differential equations from part (a) indicate:

(i) intervals on the x-axis where the solutions are increasing or decreasing;

(ii) intervals on the x-axis where the solutions are concave up or down;

(iii) any symmetries of the solutions;

(iv) any singularities of the solutions.

(c) For each of the differential equations of part (a) use MAPLE to plot the slope fields and family of solutions (these can be plotted on the same graph). Choose appropriate ranges for the x- and y-values of your plots. Each graph should contain at least 5 members of the family of solutions and one of them must satisfy the given initial conditions. Indicate on the graph which curve corresponds to this solution.

2. For each of the autonomous differential equations given below find the analytic family of solutions which satisfies the differential equation. Also find the explicit solution which satisfies the initial conditions indicated:

(i) y = 5 when x = 0

(ii) y = 0 when x = 0

(iii) y = 3 when x = 1

(iv) y = -1 when x = -2

(b) For each of the differential equations from part (a) indicate:

(i) intervals on the y-axis where the solutions are increasing or decreasing;

(ii) intervals on the y-axis where the solutions are concave up or down;

(iii) any symmetries of the solutions;

(iv) any singularities of the solutions including vertical asymptotes;

(v) any equilibrium solutions and whether they are stable or unstable.

(c) For each of the differential equations of part (a) use MAPLE to plot the slope fields and family of solutions (these can be plotted on the same graph). Choose appropriate ranges for the x- and y-values of your plots. Each graph should contain sufficient members of the family of solutions to show the various behaviour of the solutions including the equilibrium solutions if they exist. One of them should must satisfy the given initial conditions. Indicate on the graph which curve corresponds to this solution.

3. We can model fish populations which includes a constant level of harvesting at a level given by the parameter h by the differential equation

(a) Find the equilibrium solutions of this differential equation for the folowing sets of parameter values. Classify each of these solutions as stable, unstable or semi-stable.

(i) a = 1, c = 1/10, h = 8/5;

(ii) a = 1, c = 1/10, h = 5/2.

(b) Use MAPLE to plot the family of solutions for these two cases. Include the equilibrium solutions and solutions above and below these. Do these plots agree with your classification of the equilibrium solutions?

(c) Also plot the family of solutions for the parameter values a = 1, c = 1/10, h = 3. What do these results imply about the long term behaviour of the fish population under these conditions?

4. Radioactive 212Ba has a half-life of 60.5 minutes. Write down the differential equation for the quantity y of this substance as a function of time t. What are the units of t in this equation? Find the analytic family of solutions for this differential equation.