Practice Exam -1
Intro. DEs
Spring 2005
1. Consider the extinction-explosion initial value problem
dP
= kP (P − M), P (0) = P0 .
dt
a. Find the critical points
b. Find the explicit solution of the equation
c. How does the behavior of P (t) as t increases when 0 < P0 < M ?
d. How does the behavior of P (t) as t increases when P0 > M ?
e. Decide whether the critical points are stable or not
2. Let I be an open interval containing 0. Assume that p(x) and
q(x) are continuous functions on I. Is it possible that both y1 = ex and
y2 = x2 are solutions of the following equation ?
y ′′ + p(x)y ′ + q(x)y = 0.
Hint: consider Wronskian.
3. yp = 3x is a particular solution of the equation
y ′′ + 4y = 12x.
Find a solution of the equation that satisfies
y ′ (0) = 7.
y(0) = 5,
4. Solve that initial value problem
y (4) = y (3) + y ′′ + y ′ + 2y
y(0) = y ′(0) = y ′′ (0) = 0,
y (3) (0) = 30.
5. Find the implicit solutions of the differential equation
(1 + yexy )dx + (2y + xexy )dy = 0.
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