Math 334 Lecture #13
3.4: Reduction of Order; Repeated Roots
Method of Reduction of Order (dAlembert). Suppose y1 is a nonzero solution
of
y + p(t)y + q (t)y = 0.
For any scalar c, the scalar multiple
Math 334 (Ordinary Dierential Equations)
Exam 1 KEY
Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.
1. If y (x) is the solution of y =
a)
e)
sin x
, y (0) = , then the va
Math 334 Lecture #15
4.2: Homogeneous nth -order ODEs with Constant Coecients
A solution of the homogeneous linear ODE
L[y ] = a0 y (n) + a1 y (n1) + + an1 y + an y = 0
has the form y = ert if and onl
Here is the details: First dividing 4 on both sides and moving 27(4 + )2 on the right hand side:
4(6 + )3
4 (6 + )3 27(4 + )2
2
27(4 + )
(1)
Then we may do a sub u = 4 + , then we get
(6 + )3 27(4 +
Here is the details: First dividing 4 on both sides and moving 27(4 + )2 on the right hand side:
4(6 + )3
4 (6 + )3 27(4 + )2
2
27(4 + )
(1)
Then we may do a sub u = 4 + , then we get
(6 + )3 27(4 +
Problem 26 Page 143
I suppose you nished part (a) and part (b). Then from part (b), you should get ym
ym =
4(6 + )3
27(4 + )2
(1)
For part (c), we need to determine the smallest value of for which ym
From part (a) and problem 18, we get
y (t) = ae
2t
3
2t
2
+ (1 + a)te 3
3
(1)
Part (b) is kind of trick. Here we dont need to take the derivative. When do you need to take the
derivative? Usually, whe
Suppose you get
x2
w=0
x1
Move the second term on the right hand side, then we get
w+
w =
x2
w
x1
divide w on both sides, you will get
w
x2
=
w
x1
and then integrate it on both sides
w
=
w
x2
x1
which
Problem 26 Page 143
I suppose you nished part (a) and part (b). Then from part (b), you should get ym
ym =
4(6 + )3
27(4 + )2
(1)
For part (c), we need to determine the smallest value of for which ym
Math 334 Lecture #26 6.4: Discontinuous Forcing Example. Use the Laplace transform to solve the IVP
y + 4y + 4y = u1 (t) u2 (t), y (0) = 0, y (0) = 0. Applying the Laplace tranform (by its rules) give
Math 334 Lecture #16 3.5,4.3: Method of Undetermined Coecients
This is also known as the method of guessing as it applies to nth -order linear nonhomogeneous ODEs with constant coecients: L[y ] = an y
Math 334 Lecture #17
3.6,4.4: Variation of Parameters
Theorem. If y1 , y2 , . . . , yn are linearly independent solutions of
L[y ] = y (n) + p1 (t)y (n1) + + pn1 (t)y + pn (t)y = 0,
then a particular
Math 334 Lecture #18 3.7: Mechanical Vibrations Mass-Spring Systems. A spring is attached to a xed support, an object is suspended
on the spring, and a dashpot is placed about the object. [Sketch pict
Math 334 Lecture #19
3.8: Periodically Forced Vibrations
Undamped Periodically Forced Motion. This occurs when = 0:
mu + ku = F0 cos t,
where F0 is the forcing amplitude and is the forcing frequency.
Math 334 Lecture #20
5.1: Power Series
Once a fundamental set of solutions of the associated homogeneous ODE for a linear
ODE with nonconstant coecients like
y + xy + 2y = g (x)
is found, then a parti
Math 334 Lecture #21
5.2: Power Series Solutions, Part I
Example. Solve the IVP
y + xy + 2y = 0,
y (0) = 4, y (0) = 1,
using a power series
an x n .
y=
n=0
[This is a mass-spring system with a varying
Math 334 Lecture #22
5.3: Power Series Solutions, Part II
Representation Principle. When can a general solution of
y + p(x)y + q (x)y = 0
be represented by power series?
If p(x) and q (x) are analytic
Math 334 Lecture #23 6.1: The Laplace Transform An Idea. Is there an invertible linear operator Lcfw_y (t) = Y (s) that tranforms an
IVP y (t) + ay (t) + by (t) = 0, y (0) = y0 , y (0) = y0 , into an
Math 334 Lecture #24 6.2: Solving IVPs with the Laplace Transform Some Properties of the Laplace Transform.
(1) The Laplace transform is linear:
Lcfw_c1 f1 (t) + c2 f2 (t) =
0
est c1 f1 (t) + c2 f2 (t
Math 334 Lecture #25 6.3: Step Functions (and other Piecewise Continuous Functions) A Prototype. The unit step function with an upward step of 1 at c 0 is
uc (t) = [Sketch graph of the unit step funct
Yes, you are almost right. But remember that for two matrixes A,B , usually, we dont have
AB = BA.
Letg go back the question 7.4 number 6. We are trying to work with
x = Ax
we already get
x=
t t2
1 2t
Math 303 (Engineering Mathematics II)
Exam 1
RED KEY
Part I: Multiple Choice. Mark the correct answer on your scantron. Each question is worth 5
points.
1. Let
y = (y 1)2 (y 2)3 .
The equilibrium solu
Math 303 (Engineering Mathematics II)
Exam 1
RED KEY
Part I: Multiple Choice. Mark the correct answer on your scantron. Each question is worth 5
points. In problems 1 to ?, match the dierential equati
Math 303 (Engineering Mathematics II)
Exam 1
RED KEY
Part I: Multiple Choice. Mark the correct answer on your scantron. Each question is worth 5
points.
1. The solution to y =
a)
x2
is
y
3y 2 2x3 = C
F
Math 303 (Engineering Math II)
Exam 3
RED KEY
Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.
1. What is the radius of convergence for
n=1
a)
0
b)
1
c)
2
d)
3
e)
4
f)
x
1
1. Which dierential equation is equivalent to the system
x =x+y
y = x + 2y ?
a)
x x=0
b) x + 3x + x = 0
c) x 2x + x = 0
d)
x +x=0
e) x 3x + x = 0
f) x + 2x x = 0
Solution: e)
2. The Wronskian of the