MTH 244/01

Instructor: M. Kulenović
Spring 2005
University of Rhode Island

Practice problems for Exam #3

1.   Using the method of  Laplace transform, find the solution of the following initial value problem:
 y' ' + 5 y' + 4 y = 2 - x2  + sin (x)
y(0) = 1,  y'(0) = -1.

2.  Using the method of  Laplace transform, find the general solution of the system of differential equations:
x ' = x + 2y + 2
y ' = -2x + 3y - e2t .

3.   Using the method of  Laplace transform, find the solution of the following initial value problem:
x ' = 4x + 2y
y ' =-3 x - y,
y(0) = 1,  y'(0) = -1.

4.  Compute the first eight terms of the power series solution Σ an  xn about the initial point for the following initial value problem:
y'' - x2 y' + 4y = 0,
y(0) = 1, y'(0) = 1.

Find the recurrence relation satisfied by an .

5.  Compute the first seven terms of the power series solution Σ an  xn about the initial point for the following initial value problem:

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

Find the recurrence relation satisfied by an .

6. Write the Euler's recursive formula for the following initial value problem:

y' -  x2y + cos(x) y2 = 0,
y(0) = 1.

Find the first three points (x0, y0), (x0, y0), (x0, y0) of the numerical solution. Take h = 0.2.

7. Write the Euler's recursive formula for the following initial value problem:

x ' = 2 x + y2  + sin(x)
y ' =  x3 + 4 y + cos(y),
x(0) = 0, y(0) = 1.

Find the first three points (x0, y0), (x0, y0), (x0, y0) of the numerical solution. Take h = 0.1.