Math 240 Fall 2002 Final Exam
1. Solve the initial value problem
2xy dy + (y2−6x2)dx = 0, y(−2) = 0.
2. Solve the initial value problem
y′′ + 4y′+ 8y= 0, y(0) = 1, y′(0) = 0.
3. Find the general solution of the differential equation
y′′ −2y′+y= 2et.
4. Find an equation of the plane through the origin parallel to the line x= 1 −t,y= 3t,z= 2 + t
and orthogonal to the plane x+y+z= 5.
5. Let Vbe the vector space of all vectors (x1, x2, x3, x4, x5) in R5such that
x1+x3= 0
x1−x2+ 2x3−x4= 0.
What is the dimension of V?
6. Let Abe the matrix
1 2 0 0
0 1 0 0
0 0 0 −1
0 0 2 2
.
Find the inverse of the matrix A.
7. A certain 4 ×4 matrix Ahas eigenvalues 0, 1, and 2. Suppose also that
A
1
0
1
0
=
2
0
2
0
,and A
0
1
1
0
=
0
2
2
0
.
(a) Determine whether there exist a diagonal matrix Dand an invertible matrix Psuch that
A=P−1DP .
(b) If your answer to (a) is positive, find the sum of all entries of D.Hint: this does not require
finding Dor P.
8. Solve the initial value problem
xy′=yln x, y(e) = e.
9. Find the solution to the system
X′=1 1
2 2X
that also satisfies
X(0) = 1
1.
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