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MATH 3740
SAMPLE QUESTIONS FOR FINAL
Name
12/12/2012
1. Find the general solution of the first order differential equation given by;
0
xy + (2x − 3)y = 4x4 .
ANSWER: y(x) = 2x3 + Cx3 e−2x
2. Verify that the differential equation (cosx + lny)dx + ( xy + ey )dy = 0 is exact,
then solve it.
ANSWER: sinx + xlny + ey = c
3. Suppose that the population P (t) of a country satisfies the differential equation
dP
= kP (200 − P ) with k constant.Its population in 1940 was 100 million
dt
and was then growing at the rate of 1 million per year. Predict this country’s
population for the year 2000
ANSWER:153.7
4. Find the equilibruim solutions of the autonomous differential equation dx
=
dt
2
x − 4x, then determine them as stable or unstable and sketch the typical
solution curves. Finally, solve the differential equation explicitly for x(t) in
terms of t.
4x0
ANSWER: x(t) = x0 +(4−x
4t
0 )e
4 3
1
3 −5
5. Let A =
and B =
find a matrix X such that AX = B
5 4
−1 −2 5
6. The square matrix A is called orthogonal if AT = A−1 . Show that the determinant of such a matrix must be either +1 or −1.
7. The matrices A and B are said to be similar provided that A = P −1 BP for some
invertible matrix P . Show that if A and B are similar then det(A) = det(B).
8. Let W = {(a, b, c, d) : a − 4b = c + 11d = 0} be a subspace of <4 . Find a basis
of W and determine its dimension.
ANSWER: The basis {(4, 1, 0, 0), (0, 0, −11, 1)} and dimW = 2.
9. Suppose that A is an n × n matrix and k is a constant. Show that the set of all
vectors x such that Ax = kx is a subspace of <n .
ANSWER: We are trying to show that W = {x ∈ <n : Ax = kx} is a subspace
of <n . So we must verify that if x1 , x2 ∈ W then (i)x1 + x2 ∈ W , and (ii)
cx1 ∈ W for any scalar c ∈ <
10. (a)Find a general solution of the second order differential equation given by
6y 00 − 7y 0 − 20y = 0.
−4x
5x
ANSWER: y(x) = Ae 2 + Be 3
(b)Determine whether f (x) = sin2 x and g(x) = 1 − cos2x are linearly dependent or independent.
ANSWER:We must look at the wronskian of f and g.
11. Find a basis for the solution space of the given homegeneous linear system given
by:
x1 + 3x2 + 4x3 − 5x4 = 0
2x1 + 6x2 + 9x3 + 5x4 = 0
12. A nonhomogeneous differential equation,a complementary solution yc and a
particular solution yp are given. Find a solution satisfying the given initial
00
0
conditions. y − 4y = 12; y(0) = 0, y (0) = 10; yc = c1 e2x + c2 e−2x ; yp = −3
ANSWER: Any solution will be in the form of y(x) = yc + yp use the initial
conditions to find c1 and c2 .
13. Transform the 5th degree of differential equation given by;
00
0
x5 + 4x4 − 3x + 11x = lnt into an equivalent system of first-order differential
equation.
14. Find the general solution of the system of first order differential equation given
by x0 = y, y 0 = −9x + 6y.
ANSWER: x(t) = (c1 + c2 t)e3t and y(t) = (3c1 + c2 + 3c2 t)e3t
5 −4
15. Let A =
. First diagonalize the matrix A as; A = P DP −1 where D is
3 −2
a diagonal matrix consists of the eigenvalues of A, and P is the matrix whose
columns consist of the eigenvector of A. Then find A5 .