Assignment 3 Due: Mon. Apr. 16, 2001
1. (a) Find the family of explicit solutions for the following linear differential equations:
(i) xy´ + 3y = 2x3/2 + x2;
(ii) y´ - 2cot(x) y = sin2(x)
(b) Apply the Existence and Uniqueness Theorem to the differential equation in part (ii). Investigate the nature of the family of solutions at points where the theorem is not satisfied.
2. Find the family of explicit solutions for the following Bernoulli differential equations:
(i) x2 y´+ 2xy - y3/2 = 0;
(ii) y´ + 2xy + xy4 = 0;
3. Show that the following differential equations are exact and find the family of implicit solutions:
(i) (2x3 + 3y) dx + (3x + y - 1) dy = 0;
(ii) (cos(y) + y cos(x)) dx + (sin(x) - x sin(y)) dy = 0;
4. An Electrical RL circuit has an inductance L = 3 henrys and a resistance R = 6 ohms. If the applied voltage E = sin(t) e-2t volts find the current I (in amperes) in the circuit if I(0) = 0.
5. Investigate the logistic equation where the overcrowding term varies as the cube of the population. Consider the particular differential equation y´ = y(4 - y2).
(a) What are the equilibrium solutions? Classify them as stable or unstable.
(b) Solve this differential equation as a Bernoulli equation. Show that your solutions have the behaviour predicted in part (a).
6. (a) Convert the following second-order differential equations to a system of first-order differential equations:
(i) d2y/dx2 = 2x2 dy/dx - 4xy3;
(ii) 2x d2y/dx2 - 2dy/dx + 4xy = sin(2x), y(0) = 1, y´(0) = -2.
(b) Convert the following first-order differential equations into a system of autonomous first-order differential equations where both x and y depend on the independent variable t;
(i) dy/dx = (x + y)/(x - y);
(ii) dy/dx = 1/(y2 - x).
7. MAPLE can be used to plot systems of differential equations. For example, if we have the system
x´ = -2y, y´ = 4x - ty
we can define this in MAPLE as
DE:=[diff(x(t),t) = -2*y(t), diff(y(t),t) = 4*x(t) - t*y(t)];
Then to plot this system we use the command (don't forget the with(DEtools) command)
We see that this is very similar to the way we plot the solutions of single first-order differential equations except that there are now two dependent variables x and y. The ranges for t , x and y are arbitrary. You can adjust them to get a good plot. You can specify any number of initial conditions but now you must give values for both x and y at the same value of t. The options are the same as before, e.g. linecolour, stepsize, etc. The only real difference is the scene parameter which specifies which variables are plotted in the two-dimensional plot. Thus scene = [x,y] plots y vs x while scene = [t,x] plots x vs t. You can also use scene = [t,y].
If the system is autonomous, i.e. does not depend explicitly on t, the direction field will also be plotted.
(a) Use MAPLE to plot solutions to the following first-order systems. Use both scene = [x,y] and scene = [t,x] to produce two plots showing different aspects of the behaviour of these systems.
(i) dx/dt = sin(x) + cos(y); dy/dt = cos(x) - sin(y);
(ii) The system obtained in question 6 (a) (i);
(iii) The system obtained inquestion 6 (b) (ii).
(b) On the orbit plot of part (a) (iii) indicate the solution of the original differential equation which satisfies the initial condtion y = 0 at x = 1. You will have to include this initial condition on your plot command. How does this plot differ from the MAPLE plot you obtained in question 1 (d) of Assignment 2?
(c) Use the Existence and Uniqueness Theorem for systems to show that the system in part (iii) has a unique solution everywhere. In your plot of x vs t in part (a) (iii) include the initial conditions y = 1 when x = 0 and y = -1 when x = 0. Why do these curves cross in the plot if the solutions are unique?
The tutorials this week will provide a review of complex numbers. You are expected to know all of the material in Appendix A4 of the text. If you are unsure of this material come to the tutorial.