URI/Spring 2004

Instructor: Prof. M. Kulenović

Due date: March 1, 2005

Kingston, 2/15/2005

The**
**objective of this project is to use *Maple*'s symbolic
manipulation capabilities to solve certain differential equations and
to use *Maple*'s graphics capabilities to plot the required
solutions of these equations.

Use *Maple*'s worksheet that
you will find in:

**1. **Consider the exact differential equation:

(x^{2 }y
+ 2 sin(2x+y) - e^{2x }y) dx + (x^{3 }/3 + sin(2x+y)
- e^{2x }/2) dy = 0.

(b) Use dsolve command to find the particular solutions of this equation that satisfies the initial conditions y(0) = 1 and y(0) = -2.

(c) Plot the obtained solutions along with the direction field.

**2. **Consider the Bernoulli's differential
equation:

x y' + 2 y
= x^{2 } cos(x^{2 }) y^{-1 }.

(b) Use dsolve command to find the particular solutions of this equation that satisfies the initial conditions y(1) = -1 and y(1) = 4.

(c) Plot the obtained solutions along with the direction field.

**3.
(a) **Use
Abel's formula

and Maple to find a second linearly independent solution of the following differential equation with the given solution:

(2
cos (x) + sin (x) ) y'' - 5 cos (x) y' + (2 cos(x) - 4 sin(x) ) y = 0

y_{1}(x) = cos(x).

y

(b) Use dsolve command to find the particular solution of this equation that satisfies the initial conditions

y(0) = 1, y'(0) = 2.