Math 2280:
PRACTICE TEST 1
This test is closed book, closed notes test. Only small non-graphing calculator is allowed.
1. Find general solutions (implicit or explicit) of the differential equations:
a.
y3
dy
= (1 + y 4 ) cos x
dx
b.
x
dy
= 2y + x3 cos x
dx
2. The given equation models logistic population with harvesting rate h.
dy
= x(4 − x) − h
dx
a.
Suppose, h = 2. Find the critical (fixed) points of the given autonomous differential
equation, draw the phase diagram, and analyze the stability of the fixed points.
b.
Determine the dependence of the number of fixed points and their stability on the
parameter h. Sketch bifurcation diagram.
3. Verify that the given differential equation is exact. Then solve it.
y
(x3 + ) dx + (y 2 + ln x) dy = 0
x
4. Determine whether the given functions are linearly dependent or linearly independent on
the real line. Jusify your answer.
a.
f (x) = ex ,
b.
f (x) = 1,
g(x) = cos x,
g(x) = sin2 x,
h(x) = sin x
h(x) = cos2 x
5. The roots of the characteristic equation of a certain differential equation are
√
√
√
√
± 2, ±i 2, ±i 2, ±i 2
a.
b.
c.
d.
What is the order of the differential equation?
Write a general solution of this homogeneous differential equation.
Find the characteristic equation.
Find this linear homogeneous constant-coefficient differential equation.
6. Solve the initial value problem
y (3) − 2y ′′ + y ′ = 0,
y(0) = 0, y ′(0) = 2, y ′′(0) = 3