MATH 202 Exam 2, Spring 2008

1 MATHEMATICS 202 SECOND SEMESTER, 2004-05 MAKEUP QUIZ II Time: 70 MINUTES. Date: April 23, 2005. Name:————————————- ID Number:——————————– Circle Problem Session Instructor: Dr. A.Lyzzaik, Dr. H. Yamani Circle Section Number: 13, 14, 15, 16 GRADE: 1. /4 2. /8 3. /8 4. /6 5. /6 6. /10 7. /8 —————————————————– Total: /50 2 1. Use the method of undetermined coefficients to find the expected form of a particular solution yp for the differential equation y′′ − 2y′ + 5y = ex sin 2x. Caution: Need not find yp explicitly. (4 points) 3 2. Use the method of variation of parameters to find the general solution of the differential equation y′′ − 2y′ + y = ex 1 + x2. (8 points) 4 3. Solve by the power series method the initial value problem (x − 1)y′′ − xy′ + y = 0, y(0) = −2, y′(0) = 6. (8 points) 5 4. Find the general solution of the Bessel-type differential equation d dx(xy′) + � x − 4 x � y = 0; x > 0. (6 points) 6 5. Use the substitution x = et to find the general solution of the differ- ential equation x3y′′′ − 3x2y′′ + 6xy′ − 6y = 3 + ln x3; x > 0. (6 points) 7 6. Find the indicial roots of the differential equation 2xy′′ − (3 + 2x)y′ − y = 0, and derive the recurrence relation of the coefficients of the series solution of the equation associated with the larger indicial root (4+6=10 points) 8 7. Answer TRUE (T) or FALSE (F) only: (a) ——- The differential equation y′′ + P(x)y′ + Q(x)y = 0, has exactly two power series solution near a regular singular point x0. (2 points) (b) ——- The differential equation y′′ + P(x)y′ + Q(x)y = 0 has no fundamental set of power series solutions near an ordinary point x0. (2 points) (c) ——- The differential equation (x3 − 2x2 − 3x)2y′′ + (x + 1)(x − 3)2(sin x)y′ − (x + 1)y = 0 has irregular singular points. (2 points) (d) ——- The interval of convergence of series solutions of the differential equation (x3 + x)y′′ + xy′ + y = 0 about the regular singular point x = 0 is ]0, 1[. (2 points)