Page 1 of 15 LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE Friday, December 14 th Course and No. 2007 MATH 2066 EL Date................................... Cours et no........................... Total no. of pages Nombre total de pages... 15........ questions..... 15...... Professor Time allowed Professeur... Dr...... F.... Colin...................... Durée de l examen.. 3... hours................. INSTRUCTIONS Answer each question in the space provided. Justify all your answers. No material is allowed. Programmable calculators are forbidden. Name Student Id. Space reserved to the marker: 1 /8 6 /8 11 /7 2 /7 7 /4 12 /8 3 /7 8 /3 13 /7 4 /7 9 /7 14 /8 5 /8 10 /3 15 /8
Friday, December 14 th 2007 MATH 2066 EL Page 2 of 15 1. a) Determine which of the following equations are linear, nonlinear, exact or separable. Determine also the order of each equation. (Please justify your answer.) 1) t dy dt + t2 y(t) = t 2, t > 0. ( ) 2) y(t) d2 y dy dt + tan(t) + 4y(t) = 0. 2 dt 3) y (x) = 2x y x + y. b) Solve the equation 1).
Friday, December 14 th 2007 MATH 2066 EL Page 3 of 15 2. The population of mosquitoes in a certain area increases at a rate proportional to the current population, and in the absence of other factors, the population triples each month. There are 300,000 mosquitoes in the area initially, and predators eat 100,000 mosquitoes each week. Determine the population of mosquitoes in the area at any time t (in month). (Hint: Determine first the rate of growth. Be cautious! This rate is not equal to 3.)
Friday, December 14 th 2007 MATH 2066 EL Page 4 of 15 3. The following equation: y (t) + 1 t y(t) = (sin t)(y(t))n, t > 0, (1) where n is a nonnegative integer, is called a Bernoulli equation after Jakob Bernoulli. (Remark: all the sub-questions are independent from each other.) a) Find the set of all points (t, y), t > 0, for which the assumptions of the theorem of existence and uniqueness are satisfied. b) Assume that n = 0. Find the general solution of the equation (1). c) Let us assume now that n = 2. Show that the substitution v(t) = y 1 (t) reduces the Bernoulli s equation to a linear equation for v(t) (assume also that y(t) 0).
Friday, December 14 th 2007 MATH 2066 EL Page 5 of 15 4. Let the following autonomous equation y = f(y) = (y 2 2y 3) sin y, π < y < π. (2) a) Find all the equilibrium solutions of the equation (2). b) Sketch the corresponding phase line, integral curves and direction field. c) Determine the stable, unstable solutions and threshold levels. In addition, if y(t) is a solution of (2) satisfying y(0) = 2, then determine lim y(t). t +
Friday, December 14 th 2007 MATH 2066 EL Page 6 of 15 5. a) Show that the differential equation x 2 y 3 + ( x + xy 2) dy dx = 0 is not exact but becomes exact when multiplied by the integrating factor µ(x, y) = 1/(xy 3 ). b) Find the general solution and the solution satisfying the initial condition y(1) = 1. (Please leave the solutions in the implicit form.)
Friday, December 14 th 2007 MATH 2066 EL Page 7 of 15 6. Determine the general solution of y (t) + 2ay (t) + y(t) = 2e at + cos(t). where a > 1.
Friday, December 14 th 2007 MATH 2066 EL Page 8 of 15 7. Let A, B, C, D be four real numbers. Show ( that f 1 (t) ) = At+B and f 2 (t) = Ct+D A B are linearly independent if and only if DET 0. C D 8. Find the general solution of the following differential equation y (3) (t) y (t) y 2y = 0. (Hint: There is a rational root.)
Friday, December 14 th 2007 MATH 2066 EL Page 9 of 15 9. Determine a suitable form for a particular solution of the following equations if the methods of undetermined coefficients is to be used. Do not evaluate the constants. Please justify your answers. a) y (t) + 4y(t) = 3 cos(2t) + 4 cos(t) + t 2. b) y 9y = cos(3t) + e 3t + te 3t 10. Determine (without solving the problem) the largest interval in which the solution of the given initial value problem is certain to exists (and to be unique). (3t π)y (t) + (cot(t) tan(t))y(t) = 2t, y(π/4) = 1. (Reminder: cot(t) = 1/ tan(t))
Friday, December 14 th 2007 MATH 2066 EL Page 10 of 15 11. a) Determine the radius of convergence of the given power series. n 2 ( 3) n (x + 1) n n=1 e n b) Determine the Taylor series about the point x 0 = 0 for the given function. f(x) = x ( ) 1 1 x = x. 1 x
Friday, December 14 th 2007 MATH 2066 EL Page 11 of 15 12. The equation y (x) 2xy (x) + λy(x) = 0, < x < +, (3) where λ is a real constant, is known as the Hermite equation. It is an important equation in mathematical physics. a) Let us try to solve the equation (3) by means of a power series about x 0 = 0. Find the recurrence relation. (Hint: 2a 2 + λa 0 = 0 and the general recurrence relation holds for n 1.) b) Let us assume that λ = 4. Find the first three terms in each of two linearly independent solutions. (Hint: you may arbitrarily set a 0 = a 1 = 1. In addition, you will receive some marks for this part even if you didn t find the correct recurrence relation in a).)
Friday, December 14 th 2007 MATH 2066 EL Page 12 of 15 13. Let us consider the following differential equation y + 1 x (1 x) y + y = 0. (4) (1 x) a) What are the radii of convergence of the Taylor series for p(x) = 1/(1 x) and for q(x) = x/(1 x) about x 0 = 0? (Hint: use 11 b)) b) Determine a lower bound for the radius of convergence of series solutions, about x 0 = 0, of the equation (4).
Friday, December 14 th 2007 MATH 2066 EL Page 13 of 15 14. Let us consider the following Euler equation: x 2 y (x) + xy (x) + y(x) = 0. (5) (Remark: the following sub-questions are independent from each other.) a) Find the general solution of the equation (5) that is valid in any interval not containing the origin. b) Prove that x 0 = 0 is a regular singular point of the previous equation. c) Let us try to solve the equation (5) by means of a power series about x 0 = 0. Find the recurrence relation. (Hint: the relation can be written as p(n)a n = 0, where p(n) is a polynomial such that p(n) 0 for all nonnegative integers n.)
Friday, December 14 th 2007 MATH 2066 EL Page 14 of 15 (Continuation of the part c)) d) From the part c), it follows that a n = 0 for all nonnegative integers n. Thus, it is impossible to solve the equation (5) by means of a power series about x 0 = 0. Please explain why. 15. a) Solve the following initial value problem: ( ) ( ) 0 3 1 x (t) = x(t), x(0) =, (6) 1 2 1 where x(t) = ( x1 (t) x 2 (t) ).
Friday, December 14 th 2007 MATH 2066 EL Page 15 of 15 (Continuation of the part a)) b) Let the following second order linear equation with constant coefficients: ay (t) + by (t) + cy(t) = 0, (7) where a, b, c are real constants with a 0. Transform the equation (7) into a system of two first order equations, x (t) = A x(t), and show that the eigenvalues of the matrix A are equal to the roots of the characteristic equation corresponding to the equation (7). Merry Christmas and Happy New Year!