2250–4 Practice Problems
October 1, 2004
√
1. (Linear Equations) Solve the linear equation 2xy 0 (x) + 7y(x) = 3x,
y(1) = 4. Expected details include the integrating factor method and all
integration steps, by hand.
2. (Linear Equations) Solve the following initial value problem
dx
+ 5x = 15,
dt
x(0) = 5
3. (Linear Equations) Solve the initial value problem
dy
+ 3y = 2x,
dx
y(0) = 1
4. (Linear Equations) Solve the initial value problem
ey y 0 = 1, y(0) = 0
5. (Logistic DE) Solve the logistic problem P 0 = (2 − 3P )P , P (0) = 100.
Graph typical solution curves, including the equilibrium solutions. In the
graphic, label the stable and unstable equilibrium solution.
6. (Logistic DE) Consider the differential equation
dP
= P 2 − 2P
dt
which could be a model for a certain population problem.
a) Find the equilibrium solutions.
b) Sketch the slope field for this differential equation. Onto the slope
field sketch graphs of the solutions to the four initial value problems with
P (0) = −1, P (0) = 0, P (0) = 1, P (0) = 2, P (0) = 3. (You don’t need
formulas for the solutions to make the sketches!)
c) Which of the equilibrium solutions are stable? Which are unstable?
e) Find an explicit solution to the initial value problem for this differential
equation, with P (0) = 3. Explain what happens to your solution as time
increases.
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7. (Logistic DE) Consider the differential equation
dP
= −P 2 + 4P − 3
dt
which models a certain population problem.
a) Find the equilibrium solutions.
b) Sketch the slope field for this differential equation. Onto the slope field
sketch graphs of the solutions to the three initial value problems with
P (0) = 0, P (0) = 2, P (0) = 4. (You don’t need formulas for the solutions
to make the sketches!)
c) Which of the equilibrium solutions are stable? Which are unstable?
d) Give a population model which leads to differential equations of this
type. Be as precise as you can, so that you can account for the signs of
all three terms on the right-hand side of the differential equation.
e) Find an explicit solution to the initial value problem for this differential
equation, with P (0) = 2. Verify that your limiting population agrees with
what your sketch predicted in part (b).
8. (Velocity and Acceleration)
(a) Solve 5v 0 = −100 − 50v, v(0) = 50.
(b) Solve y 0 (t) = v(t), y(0) = 10, where v(t) is the answer from (a).
(c) Find the terminal velocity v∞ = lim v(t).
t→∞
9. (Velocity and Acceleration)
(a) Solve v 0 = −10 − 5v, v(0) = 0.
(b) Solve y 0 (t) = v(t), y(0) = 10, where v(t) is the answer from (a).
(c) Find the terminal velocity v∞ = lim v(t).
t→∞
10. (Velocity and Acceleration) Consider a body that moves horizontally
through a medium whose resistance is proportional to the square of the
velocity v, so that dv/dt = −kv 2 . Show that
v(t) =
and that
x(t) = x0 +
v0
1 + v0 kt
1
ln(1 + v0 kt).
k
To be continued . . .
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