MATH 202 Exam 2, Spring 2008

MATHEMATICS 202 SPRING SEMESTER 2005-06 QUIZ II Time: 70 MINUTES. Date: April 8, 2006. Name:—————————————————————————- ID Number:———————————————————————– Section Number:—————————————————————— Course Instructors: Prof. Abdallah Lyzzaik and Dr. Hassan Yamani Question Grade 1 /16 2 /16 3 /16 4 /16 5 /16 6 /20 TOTAL /100 Answer The Following Six Questions On The Page Allocated For Each Question (You May Use The Back Of The Pages If Needed). 1 2 1. Find the general solution of the differential equation (16 points) x(x + 1) y′′ + (2 − x2) y′ − (x + 2) y = (x + 1)2 whose complementary solution is y = c1ex + c2 x−1 for all real x ̸= 0. 3 2. Find the general solution of the differential equation (16 points) y′′ − 2y′ + y = x ex. 4 3. Suppose m1 = 1 and m2 = 1 + √ 3 i are the characteristic roots each of multiplicity 2 of the characteristic equation of a homogeneous linear differential equation. Write down the general solution of the differential equation if it is (a) an equation of constant coefficients. (8 points) (b) a Cauchy-Euler differential equation. (8 points) 5 4. Classify each singular point of the differential equation x4(x2 − 36)(x − 3)2y′′ + x(1 − cos x)y′ + 2x2y = 0 as regular or irregular, and find the least possible radius of convergence for the series solution of the differential equation at each regular point. Justify your answers. (16 points) 6 5. Use the substitution x = et to transform the Cauchy-Euler differ- ential equation 2x2y′′ + 5xy′ − 2y = 2 ln x, (x > 0), to a differential equation with constant coefficients, then solve the orig- inal equation. (16 points) 7 6. Show that the indicial roots of the differential equation 2x2y′′ + (x2 − x)y′ + y = 0, at the regular singular point x = 0 are 1/2 and 1, and find the solution of the differential equation associated with the root 1/2. (20 points)