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 x3 ).
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.