Assignment 2 Due: Friday, Mar. 30, 2001
1. (a) Use the Existence and Uniqueness Theorem to determine where the following differential equations have unique solutions:
(b) Find the regions of the x-y plane where the solutions to the differential equations are monotonically increasing or decreasing. Also find any equilibrium solutions.
(c) Find a solution to the differential equation (i) with the initial condition y(0) = 1 that is valid for all x. Be careful to check that you have the correct solution.
(d) Use MAPLE to plot solutions of the differential equation (ii) with initial conditions of the form y(0) = y0. Chose at least one value of y0 in each of the following intervals: 1 < y0; -1 < y0 < 1; y0 < -1. Do you think the curves that MAPLE produces are correct? Explain what is going on.
2. (a) Find the regions of the x-y plane where the solutions to the following differential equation are monotonically increasing or decreasing. Derive any symmetries that these differential equations possess and find any equilibrium solutions.
(b) Use MAPLE to plot several solution curves for each of these differential equations. Choose initial values to illustrate the properties derived in part (a).
(c) Use the comparison theorem to show that the solution of the differential equation given in (i) with initial condition y(0) = 2 has a vertical asymptote in the interval 0.5 < x < 0.55 by noting that for
0 < x < 1, y2 - 1 < y2 - x2 < y2.
3. (a) Find the analytic family of solutions to the following separable differential equation:
What symmetries does this differential equation have? What are the equilibrium solutions? Use MAPLE to plot several members of this family that illustrate these properties.
(b) A chemical reaction depends on the amount of the reagent Y and the temperature of the reagent. Initially there is 10 kg of the reagent in solution and the temperature is 300 K. If the reagent decreases at a rate of 0.1% of the reagent present times the temperature of the solution per minute and the solution is heated at the rate of 2 K per minute, find the amount of reagent in the solution as a function of time.
4. Find analytic families of solutions for the following homogeneous differential equations. Also find all solutions of the form y = mx for these differential equations where m is a constant.
5. (a) Find the orthogonal trajectories of the family of curves given by the following equations where p is a paramter which takes on different values:
i) y = pe-x ii) x2 + py2 = 1
(b) Use MAPLE to plot these family of curves and their orthogonal trajectories in the first quadrant by choosing various values of p and the constant of integration in the orthogonal trajectories. Choose the same ranges for x and y so that the angles between curves appear correctly on the graph.
Notes on MAPLE:
You can use MAPLE to make plots. For example, to plot the curves y = x2, y = sin(x) and y = ln(x) in the region a < x < b, c < y < d on the same graph use the command
plot([x2,sin(x),ln(x)],x = a..b,y=c..d,colour=black);
Some students ran into problems saving MAPLE files because of their large size. To avoid this problem use the `Remove output -> worksheet' command on the Edit menu to remove the plots before saving the file. You can regenerate the plots when you open the file by using the `Execute -> worksheet' command which is also on the Edit menu. You can also temporarily save files on the C: disk but these will automatically be deleted on logout.
The printer in the Jupiter lab can be unreliable. Don't forget that you need to bring your own paper. You can also print out your assignment in Steacie by choosing this printer from the print dialogue box. The Steacie printer is more reliable but costs more.
First Term Test - Wednesday, March 28