Exam 2 - Name: Exam 2 Instructions. Answer each of the...

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Unformatted text preview: Name: Exam 2 Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without supporting work. Put your name on each page of your paper. 1. [18 Points] Solve the initial value problem 2 x 2 y 00 + xy- y = 0; y (1) = 1 , y (1) =- 1 . I Solution. This is an Euler equation, so we first find the indicial polynomial q ( r ) = 2 r ( r- 1) + r- 1 = 2 r 2- r- 1 = (2 r + 1)( r- 1). The roots are r 1 = 1 and r 2 =- 1 / 2 so the general solution has the form y = c 1 x + c 2 x- 1 / 2 and the coefficients c 1 and c 2 are determined from the initial conditions: 1 = y (1) = c 1 + c 2- 1 = y (1) = c 1- 1 2 c 2 . Subtracting the second equation from the first gives 2 = (3 / 2) c 2 so that c 2 = 4 / 3 and substituting in the first equation gives c 1 =- 1 / 3. Hence y =- 1 3 x + 4 3 x- 1 / 2 . J 2. [18 Points] Solve the differential equation x 2 y 00 + 3 xy + y = x- 1 ( x > 0) . You may find the integral formula R x k log xdx = ( x k +1 k +1 log x- x k +1 ( k +1) 2 + C if k 6 =- 1 1 2 (log x ) 2 if k =- 1 of use. I Solution. This is a nonhomogeneous equation so we will use variation of parame- ters. First solve the associated homogeneous equation, which is an Euler equation, by finding the indicial polynomial q ( r ) = r ( r- 1) + 3 r + 1 = r 2 + 2 r + 1 = ( r + 1) 2 , which has a single root r 1 =- 1 of multiplicity 2. Hence, the associated homogeneous equation has the solution y h = c 1 x- 1 + c 2 x- 1 log x....
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This note was uploaded on 03/20/2011 for the course MATH 4027 taught by Professor Adkins during the Spring '06 term at LSU.

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Exam 2 - Name: Exam 2 Instructions. Answer each of the...

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