University of Toronto at Scarborough

University of Toronto at Scarborough Department of CMS, Mathematics MAT B44F 2015/16 Problem Set #3 Due date: in tutorial, week of Nov 16, 2015 Do the following problems from Boyce-Di Prima. S. 3.5: 7, 9 (9th ed: 5,7) S. 3.6: 6, 10, 13, 14, 16 S. 5.2 #7, 10 S. 5.3 #11 S. 5.4 #39 1. Find a particular solution yp of each of the following equations. (a) y 00 + 16y = e 3x (b) y 00 - y 0 - 6y = 2 sin 3x (c) y 00 + 2y 0 - 3y = 1 + xex (d) y 00 + y = sin x + x cos x 2. Use the method of variation of parameters to find a particular solution of the following differential equations. (a) y 00 + 9y = 2 sec 3x (b) y 00 - 2y 0 + y = x -2 e x (c) x 2 y 00 - 3xy0 + 4y = x 4 (d) x 2 y 00 + xy0 + y = ln(x) 3. Use the method of undetermined coefficients to find particular solutions of the following equations: (a) y 00 + 9y = 4 cos 3x (b) y 00 + 4y 0 + 4y = 3e -2x + e -x 4. For x > 0, find the general solution of the equation 2x 2 y 00 + xy0 - y = 3x - 5x 2 . 1 5. Use series methods to solve the differential equation y 00 + xy = 0. 6. Solve the following initial value problem using power series. First make a substitution of the form t = x - a, then find a solution P n cnt n of the transformed differential equation: (2x - x 2 )y 00 - 6(x - 1)y 0 - 4y = 0; y(1) = 0, y0 (1) = 1. 7. Consider the equation y 00 + xy0 + y = 0. (a) Find its general solution in terms of two power series y1, y2 in x, where y1(0) = 1 and y2(0) = 0. (b) Use the ratio test to verify that the series y1 and y2 converge for all x. (c) Show that y1 is the series expansion of e -x 2/2 . Use this fact to find a second linearly independent solution by the method of reduction of order. 8. Determine whether x = 0 is an ordinary point, a regular singular point, or an irregular singular point. If it is a regular singular point, find the exponents of the differential equation at x = 0. (a) xy00 + (x - x 3 )y 0 + (sin x)y = 0 (b) x 2 y 00 + (cos x)y 0 + xy = 0 (c) x(1 + x)y 00 + 2y 0 + 3xy = 0 9. Solve the following differential equation by power series methods (the method of Frobenius): 2x 2 y 00 + xy0 - (1 + 2x 2 )y = 0 10. Solve the following differential equation by power series methods (the method of Frobenius): 2xy00 - y 0 - y = 0 2