MTH 244/01

Instructor: M. Kulenovic
Spring 2005
University of Rhode Island

Practice problems for Exam #2

1.   Using the method of undetermined coefficients, find the solution of the following initial value problem:
 y' ' - y' + 2 y = e2x - x2  + sin (x)
y(0) = 1,  y'(0) = -2.

2.   Using the method of undetermined coefficients, find the general solution of the following ODE:
y' ' - 4 y' + 4 y = e2x + 5 x2 . 

3.   Using the method of undetermined coefficients, find the solution of the following initial value problem:

x2 y''  + 5 x y' + 29 y = 2 x2 + x 
y(1) = -2,  y'(1) = 1.

4.  
(a)   Use the Abel's formula to find a second linearly independent solution of the following differential equation with the given solution:
x2 (ln(x) - 1) y''  - x y' +  y = 0
y1(x) = x.

(b)   Find the general solution of the following equation:  x2 (ln(x) - 1) y''  - x y' +  y = x. Use the variation of parameters formula.

5.  Using the elimination method to find the general solution of the system of differential equations:
x ' = x + 2y + 2
y ' = -2x + 3y - e2t .

6.   Find the general solution of the system of differential equations by the matrix method:
x ' = 4x - 2y
y ' = x + y.

7.  Use the variation of parameters formula to find the particular solution of the following system:
x ' = 4x + 2y + 4
y ' = -3 x - y.-3.