Page 1 of 15 LAURENTIAN UNIVERSITY UNIVERSITÉ LAURENTIENNE Wednesday, December 13 th Course and No. 2006 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 /8 2 /7 7 /4 12 /7 3 /7 8 /3 13 /8 4 /7 9 /7 14 /8 5 /8 10 /7 15 /3
Wednesday, December 13 th 2006 MATH 2066 EL Page 2 of 15 1. a) Determine which of the following equations are linear, nonlinear or separable. (Please justify your answer.) ( ) 2 1) t 2 d2 y dy dt + cos t + 2y(t) = 0 2 dt 2) dy dt + et y(t) = t 2 3) y (x) = x2 cos y b) Solve the equation 3) (Please leave the solution in the implicit formulation.)
Wednesday, December 13 th 2006 MATH 2066 EL Page 3 of 15 2. Let s define the double-life of an investment as the time required for an amount of money deposited in a bank account to increase to the double of its original value. Let r be the interest rate and τ, the double-life. a) Find the solution of the initial value problem Q (t) = rq(t) Q(0) = Q 0. b) Show that the double-life τ and the interest rate r satisfy the equation rτ = ln 2.
Wednesday, December 13 th 2006 MATH 2066 EL Page 4 of 15 3. Using the method of integrating factors, we will find the solution of the following initial value problem: y (t) + 2y(t) = e t, y(0) = 1. a) The integrating factor, denoted µ(t), must satisfy the equation: µ (t) = 2µ(t). (1) Find a suitable integrating factor µ(t). b) Multiply the original equation by µ(t), use the relation (1) and find the solution of the previous initial value problem.
Wednesday, December 13 th 2006 MATH 2066 EL Page 5 of 15 4. Let the following autonomous equation y = f(y) = y 3 2y 2 y + 2. (2) a) Find all the equilibrium solutions of the equation (1). (Hint: f(2) = 0.) b) Sketch the corresponding phase line, integral curves and direction field. c) Determine the stable, unstable solutions and threshold levels.
Wednesday, December 13 th 2006 MATH 2066 EL Page 6 of 15 5. a) Show that the differential equation 2xy 2 y sin(xy) + ( 2x 2 y x sin(xy) + cos(y) ) dy dx = 0 is exact. ( π ) b) Find the general solution and the solution satisfying the initial condition y = 2 0. (Please leave the solutions in the implicit form.)
Wednesday, December 13 th 2006 MATH 2066 EL Page 7 of 15 6. a) Determine the general solution of y λ 2 y = ae t where λ > 0 and λ 1 and where a is a real number different from zero. b) Use the method of variation of parameters to find a particular solution of the preceding differential equation.
Wednesday, December 13 th 2006 MATH 2066 EL Page 8 of 15 7. Determine whether the given set of functions is linearly dependent or linearly independent. f 1 (t) = t 1, f 2 (t) = t 2 + 1, f 3 (t) = t 2 1. 8. Find the general solution of the following differential equation y (3) 2y + 4y 8y = 0 given that r 1 = 2 is an obvious root of the characteristic equation. (Hint: use the Euclidian algorithm for polynomials.)
Wednesday, December 13 th 2006 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 (3) 3y + 3y y = e t (Hint: r 1 = 1 is a triple root of the characteristic equation.) b) y + 4y = cos(2t) c) y + 4y = e 2t
Wednesday, December 13 th 2006 MATH 2066 EL Page 10 of 15 10. a) Determine the radius of convergence of the given power series. (x x 0 ) n 2 n n=1 b) Determine the Taylor series about the point x 0 = 1 for the given function. f(x) = 1 x
Wednesday, December 13 th 2006 MATH 2066 EL Page 11 of 15 11. We want to solve the following differential equation by means of a power series about the point x 0 = 0. (1 x 2 )y + 2y = 0 a) Find the recurrence relation. (Hint: 2a 2 + 2a 0 = 0, 6a 3 + 2a 1 = 0 and the general recurrence relation holds for n 2.) b) 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 problem even if you didn t find the correct recurrence relation in a).)
Wednesday, December 13 th 2006 MATH 2066 EL Page 12 of 15 12. Let us consider the following differential equation y + 1 x y + 1 y = 0. (3) ( 1 2x) a) What are the radii of convergence of the Taylor series for p(x) = 1/x and for ( q(x) = 1/( 1 2x) about x 0 = 1? ) 1 Hint: use 10 b) and observe that = 1 ( 1 2x) (1 2(x+1)) b) Determine a lower bound for the radius of convergence of series solutions, about x 0 = 1, of the equation (3).
Wednesday, December 13 th 2006 MATH 2066 EL Page 13 of 15 13. a) Find the general solution of the Euler equation that is valid in any interval not containing the origin: x 2 y (x) + 3xy (x) + y(x) = 0. b) Prove that x 0 = 0 is a regular singular point of the previous equation. c) Using the trial function y p = Ax + B, where A and B are real constants, find a particular solution of the equation: x 2 y (x) + 3xy (x) + y(x) = x + 1. (4) (Hint: this is the method of undetermined coefficients for the Euler equations.)
Wednesday, December 13 th 2006 MATH 2066 EL Page 14 of 15 d) Deduce from a) and c) the form of the general solution of the preceding equation (4). 14. a) Find the general solution of the following system: [ ] 4 3 x (t) = x(t), (5) 8 6 where x(t) = ( x1 (t) x 2 (t) ).
Wednesday, December 13 th 2006 MATH 2066 EL Page 15 of 15 b) An equilibrium solution of a system of first order equations is simply a constant solution (i.e. x (t) = 0 or x (t) = 0). For instance, x(t) = 0 (or x (t) = 0) is always an equilibrium solution of the system x (t) = Ax(t) since A 0 = 0. Find all the equilibrium solutions of the previous system. (Hint: in this case, there is an infinite number of equilibrium solutions.) 15. Determine (without solving the problem) the largest interval containing the point t 0 = 1 in which the solution of the given system of first order equations is certain to exists (and to be unique). x 1 (t) = ln(t)x 1 (t) + cos(t)x 2 (t) + tan(t) x 2 (t) = t 2 x 1 (t) + tx 2 (t) + 1/t. Merry Christmas and Happy New Year!