# MTH 244/01

Spring 2005
University of Rhode Island

Practice problems for Exam #1

1.   Solve the following initial value problem for the differential equation:

 cos(x) y' - sin(x) y = 1 y(0) = 1.
2.     Find the general solution of the homogeneous differential equation:    dy/dx = y ( y2 + 3 x2 ) / ( 2 x ).

3.     Find the general solution of the exact differential equation:  (2x - y sin (x) + 2 e2x+y )dx + ( cos x + e2x+y )dy=0 .

4.     Solve the following initial value problem for the Bernoulli differential equation:

 y' - x y = x y2 y(0) = -2.

5.     Solve the following initial value problem:

 x2 y''  + 5 x y' + 29 y = 0 y(1) = -2,  y'(1) = 1.

6.
(a)   Use the method of reduction of order or 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.
Find the particular solution of this equation that satisfies the initial condition  y(1) =1, y'(1) = 2

(b)   Can you justify the uniqueness of the solution of this IVP without finding the general solution of the above equation ? Explain.