Name:
Section Number:
110.302 Dierential Equations
FALL 2012
MIDTERM EXAMINATION B SOLUTIONS
October 12, 2012
Instructions: The exam is 7 pages long, including this title page. The number of points each
problem is worth is listed after the problem number.
THEORY: SEPARABLE DIFFERENTIAL EQUATIONS
110.302 DIFFERENTIAL EQUATIONS
PROFESSOR RICHARD BROWN
A rst-order dierential equation of the form
y = f (t, y )
is called separable if the function f can be written
f (t, y ) = g (t)h(y ),
for two other functions
Lecture 25: 7.8 Repeated eigenvalues. Recall rst that if A is a 2 2 matrix
and the characteristic polynomial have two distinct roots r1 = r2 then the corresponding eigenvectors x(1) and x(2) are linearly independent for if c1 x(1) +c2 x(2) = 0
it follows
Lecture 24: 7.6 Complex eigenvalues.
Ex Find the solution to the system
]
[ ]
[
[ ]
x1
1 2
a
x = Ax,
where x =
, A=
,
x(0) =
,
x2
2 1
b
Sol 1 First we want to nd the eigenvalues r and eigenvectors = 0:
(7.6.1)
A = r
(A rI) = 0.
The eigenvalues are solutio
Lecture 5: 2.3 More models.
Model III: Mixing of chemicals. Suppose a time t = 0 a tank contains Q0 lb
of salt dissolved in 100 gal of water. Assume that water containing 1 lb of salt/gal
4
is entering the tank at a rate of r gal/min, and that the well-st
Problem 5. 20 points Let X ~Norm({l,42). X "
H
NJ
aEmPQ<X<®. H
['0 walw mm
-\
(7&4 ~23)" PLE 5
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b. If Y~Norm(4, 22), would you expect the P(2 < Y < 6) to be larger or smaller than the probability you
found in part a?
550.310 Exam I
Name: R E; H) H
Section Number (or TA name): A r S ackmm
I certify that I did not give or receive any unauthorized aid on this exam.
Signature; 4 r 2 i
Problem 1. 20 pts. Consider a jar with four balls in it. Two of the bails are red and
Problem 5. {20 points) Suppose X1,X2, . . .,X190 ~ Norm(,u,25) are iid random variables. The infor~
mation contained in the sample about the unknown p is 13350 : 4.
a. Show X is an unbiased estimator of Jun
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6. Use proof by contradiction to show that is irrational.
Suggestion: Assume the rst fraction you use in your proof has been reduced to lowest
terms.
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NAME:
Problem 1:
Problem 2:
Problem 3:
Problem 4:
Problem 5:
Problem 6:
Total:
MATH 302
October 6, 2014
No Calculators or Notes or Books.
MIDTERM I version A Fall 14
Simplify your answer as much as possible.
)
SECTION: ; d2 1. (a) Solve the initial value
Lecture 26: 7.7 The exponential matrix.
We will now nd a nice way to the express the solution to the system
(7.7.1)
x = Ax,
where A is a 22 matrix, analogous to the formula for the solution of one equation.
We can nd two solutions to (7.7.1); x1 (t) and x
Lecture 27: 7.9 Nonhomogeneous equations. There are several methods in
the book but we will only go over using diagonalization and the exponential matrix.
Diagonalization Suppose that A is an nn matrix with n linearly independent
eigenvectors A(k) = k (k)
Lecture 22 7.3 Eigenvectors. The equation
(7.3.1)
Ax = y
or
a11 x1 + a12 x2 = y1
a21 x1 + a22 x2 = y2
can be viewed as a map transforming the vector x R2 into the vector y R2 .
Examples of such transformations are scalar multiplication or rotations of a v
EXAMPLE: ALMOST LINEAR SYSTEMS: THE PENDULUM
110.302 DIFFERENTIAL EQUATIONS
PROFESSOR RICHARD BROWN
In class, we wrote out the ODE for the pendulum; it is a second-order, nonlinear, autonomous, ODE with constant coecients, given as
+ + 2 sin = 0.
Written
EXAMPLE: THE WRONSKIAN DETERMINANT OF A
SECOND-ORDER, LINEAR HOMOGENEOUS DIFFERENTIAL
EQUATION
110.302 DIFFERENTIAL EQUATIONS
PROFESSOR RICHARD BROWN
Problem. Solve the ODE 2y + 8y 10y = 0.
Strategy. Solving this ODE means nding a fundamental set of solut
EXAMPLE: THE WRONSKIAN DETERMINANT OF A
SECOND-ORDER, LINEAR HOMOGENEOUS DIFFERENTIAL
EQUATION
110.302 DIFFERENTIAL EQUATIONS
PROFESSOR RICHARD BROWN
Given a second order, linear, homogeneous dierential equation
y + p(t)y + q (t)y = 0,
where both p(t) and
Name:
Section Number:
110.302 Dierential Equations
FALL 2012
MIDTERM EXAMINATION A SOLUTIONS
October 12, 2012
Instructions: The exam is 7 pages long, including this title page. The number of points each
problem is worth is listed after the problem number.
Lecture 32: 9.6 Energy and Liapunov functions.
Energy conservation for the Pendulum. Consider the pendulum again:
+ + 2 sin = 0,
2 = g/L
With x = and y = we get the system
x = y,
y = y 2 sin x,
2 = g/L
The critical points are y = 0 and sin x = 0, i.e.
Lecture 13: 3.6: Nonhomogeneous equation, variation of parameters.
We will now give a general method for nding particular solutions for second order
linear dierential equations that in principle works for any nonhomogeneous term.
Let us rst illustrate the
Lecture 23: 7.5 Linear systems of dierential equations.
Ex 2 Find the solution to the system
[ ]
[
]
[ ]
x1
6 2
a
x = Ax,
where x =
, A=
,
x(0) =
,
x2
2 9
b
Sol First we want to nd the eigenvalues r and eigenvectors = 0:
(7.5.5)
A = r
(A rI) = 0.
The eige
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Ax(BC)=(A><B)rI(A><C).
3. Let R1 and R2 be equivalence relations on a set S. Prove that R1 0 R2 is also an
equivalence relation. (Remember: relations are sets of ordered pairs.)
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