Engineering Mathematics - MATH 3321 - Spring 2005
Practise for Exam1
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Exercise 1. (20 points) Find the general solution of the equation
xy0+ 2y= 6xex2
(assume x6= 0).
Exercise 2. (20 points) Find the solution of the initial value problem
(x2+ 6)y0= 2xy
y(2) = 20
Exercise 3. (20 points) Consider the following linear system:
x1+ax2−2x3= 3
2x1+x2+x3= 3
3x1+x2+x3= 6
where ais a real number.
1. Use the property of the matrix of the coefficient to determine for which values of athe
system above has ONLY ONE SOLUTION;
2. find the solution of the system assuming a= 1 (THIS IS A PRACTICE FOR THE
EXAM. TRY TO SOLVE THE SYSTEM USING BOTH THE ROW REDUCTION
AND THE CRAMER’S RULE BECAUSE YOU SHOULD KNOW BOTH FOR THE
EXAM!!!);
3. (optional: 10 extra points ) what happens for a=−2?
Exercise 4. (20 points) Consider the following matrices:
A=·1−1
1 1 ¸B=·−122
2 0 3 ¸.
Compute the following operations, if possible:
(i)A−2B(iv)BB T
(ii)−B(v)A0B
(iii)AB0(vi)BTA
Think what differences there would be if Ais defined as follows:
A=·1−1
−1 1 ¸
1
