Math 251 (FALL 2014)
Review Version 1
Fall 2014
Exam I
1. (40 points) Solve (a) y
dy
+ 8 sin (πx) = 0
dx
2. (20 points) Solve the following IVP:
(b)
dy
x
=− ,
dx
4y
dy
x dx
+ 2y = 3x − 5,
y(0) = −1
y(1) = 6
3. (20 points) Find the general solution to (x + y)2 dx + (2xy + x2 − 1)dy = 0
4. (20 points) Solve the following Equation using y = vx:
(y 2 − xy) dx + x2 dy = 0
Exam II
1. (15 points) Consider the RL circuit with a 5-ohm resistor and a 0.5-H inductor carries a
current of 2 A at t = 0, at which time a voltage source E(t) = 3 cos 120t V is added.
(a) Write down the initial value problem of the inductor current I for this circuit.
(b) Supposed you are given that the particular solution to your IVP above is
I = I(t) =
3
1447 −10t
(12 sin 120t + cos 120t) +
e
,
725
725
determine the inductor voltage at any time t.
2. (20 points) A brine solution of salt flows at a constant rate of 8L/min into a large tank
that initially held 100L of brine solution in which was dissolved 0.5kg of salt. The solution
inside the tank is kept well stirred and flows out of the tank at the same rate. Assume the
concentration of salt in the brine entering the tank is 0.05kg /L.
(a) Write down the initial value problem (IVP) of the population and
(b) Write down the particular solution of this IVP.
(c) When will the concentration of salt in the tank reach 0.2kg /L?
3. (15 points) Assuming the Logistic Population Model of the US population. Using data
point in 1900 with 76.21 million (i.e. t = 0 here), 1950 with 151.33 million and 2000 with
281.42 million people, we obtain A = 0.00001295941 and p1 = 1168.596
(a) Write down the particular solution of this IVP.
(b) What is the prediction of the US population in the year 2014?
4. (15 points) In 1950, the world’s population was 2,555,982,611. With a growth rate of approximately 1.68% (i.e. the k value in the Malthusian Population Model is 0.0168).
(a) Write down the initial value problem (IVP) of the population.
(b) Write down the particular solution of this IVP.
(c) What was the population in 1960?
(d) At this rate when will the world reach 7 billion?
5. (15 points) A cold beer initially at 32◦ F warms up to 42◦ F in 5 min while sitting in a room
of temperature 65◦ F . How warm will the beer be if left out for 20 min?
Math 251 (FALL 2014)
Review Version 1
Fall 2014
6. (20 points) A big heavy iron ball of mass 50 kg is release from rest 100 m above the ground
and allowed to fall under the influence of gravity. Assuming the force due to air-resistance
is proportional to v with proportionality constant b = 50 N-sec/m.
(a) Write down the IVP for the velocity v of the iron ball.
(b) Write down the particular solution v(t) at any time t.
(c) Write down the equation of motion of the iron ball.
(d) Determine when the iron ball will hit the ground.
Exam III
1. (15 points) Given the non-homogenous DE y 00 + 2y 0 − 3y = f (t) has complementary solution
given by yh (t) = c1 e−3t + c2 et . Suppose the non-homogenous part f (t) is given below, write
down the form of particular solution (you do not need to find the constants)
(a) f (t) = 7 cos t
(c) f (t) = (2t2 + 3t) cos 3t
(b) f (t) = 7 sin 2t
(d) f (t) = (2t2 + 3t)e−3t
(e) f (t) = 2tet sin t − et cos t
2. (20 points) Find the general solution of the following (homogeneous) DE
(a) y 00 + 3y 0 + 2y = 0
(b)
y 00 − 4y 0 + 4y = 0
3. (20 points) Given that the homogeneous (or complementary) solution of y 00 + 4y = 4t2 is
given by yh (t) = c1 cos 2t + c2 sin 2t for arbitrary constants c1 and c2 .
(a) Using the method of undetermined coefficients, find its particular solution
(b) Write down its general solution.
(c) Suppose y(0) = 0 and y 0 (0) = 2. Find the unique solution to the IVP.
4. (15 points) Find the Wronkians of the following sets of functions. Then decide whether it
is linearly independent or linearly dependent.
(a)
{2e3t , te3t }
(b)
{2, cos t, sin t}
5. (20 points) Given the nonhomogenous differential equation: y 00 + 9y = 3 tan 3t. and given
that yh (t) = c1 cos 3t + c2 sin 3t solves the corresponding homogeneous DE. Using variation
of parameters, find a particular solution to the given DE.
6. (10 points) TRUE or FALSE
(a) T
F
Knowing one non-zero solution to y 00 + p(t)y 0 + q(t)y = 0, we can find
another solution using reduction of order.
(b) T
F
(c) T
F
The reduction of order method is also good for finding a particular solution to a nonhomogeneous differential equation.
For a homogeneous differential equation, we use the variation of parameters to find its particular solution.
(d) T
F
(e) T
F
If yh (t) = c1 y1 + c2 y2 is the complementary solution and yp (t) is a
particular solution, then yh (t) + yp (t) is known as the general solution
to the nonhomogeneous DE.
If y1 (t) and y2 (t) are linearly independent, then the Wronskian W (t)
will always be zero.
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Math 251 (FALL 2014)
Review Version 1
Fall 2014
Exam IV
function
1
eαt
sin bt
f (t) + g(t)
cf (t)
tn f (t)
eαt f (t)
f 0 (t)
Laplace Transform
1/s
1/(s − α)
b/(s2 + b2 )
F (s) + G(s)
cF (s)
(−1)n F (n) (s)
F (s − α)
sF (s) − f (0)
1. (20 points) Find the Wronskians of the following sets of functions. Then decide whether
they are linearly independent or linearly dependent.
(a)
{1, xe2x , 2e2x }
(b)
{3 sin t + cos t, cos t, sin t}
2. (15 points) Given the nonhomogeneous DE: y 000 − 3y 00 + 4y = e2x + xe−x + sin 2x
(a) Write down an annihilator for the nonhomogeneous part.
(b) Write down an operator L such that the general solution L[y] = 0 has the same general
solution as the above nonhomogeneous DE.
(c) What is the general solution (there is no need to solve the constants).
(d) Which constants can be determined if you substitute your general solution into the
above nonhomogeneous DE ?
3. (15 points) Write down a linear operator (with constant coefficients) that annihilates each
of the following functions
(a) f (t) = 7 cos 2t
(b) f (t) = 7t sin t
(d) f (t) = (2t2 + 3t)e−3t
(c) f (t) = (2t2 + 3t) cos 3t
(e) f (t) = et cos t
4. (20 points) Given that r3 + 2r2 − 9r − 18 = (r2 − 9)(r + 2) and given that the particular
solution to y 000 + 2y 00 − 9y 0 − 18y = 22 − 18x2 − 18x is yp (x) = x2 − 1.
(a) Write down the general solution
(b) Find the unique solution that satisfies y(0) = −2, y 0 (0) = −8, y 00 (0) = −12.
5. (20 points) Given the 3rd order DE, y 000 + y 0 = tan x has fundamental set of solution given
by {1, cos x, sin x}. Supposed you assume that yp (x) = v1 (x) + v2 (x) cos x + v3 (x) sin x.
Z
Given that v1 (x) =
tan x dx, find v2 (x) and v3 (x).
6. (10 points) Using only the table provided (above), find the Laplace Transform of
(a) et t sin 2t
(b) cos 3t + e6t − 5t2
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