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Math 3310
Section 4.5 Solutions
Ryan Livingston
Section 4.5 Page 187: 3-29 odd
For Problems 3-8, Find the General Solution
3.
y 00 + y 0 = 1; yp (t) = t
SOLUTION:
We rst nd the homogeneous solution, so we dene y = e
2
t
+
=
and get
1
= 0 and
2
=
0
=
( + 1
Math 3310
Section 4.7 Solutions
Ryan Livingston
Section 4.7 Page 200
For Problems 9-14, Find the General Solution to the Cauchy Euler Equation
9.
t2 y + 2ty 6y = 0
SOLUTION:
We suppose that y = tr and y = rtr1 and y = r (r 1) tr2 . If we make these substi
Math 3310
Section 5.2 Solutions
Ryan Livingston
Section 5.2 Page 250: 3-17 odd
For problems 3-18, use the elimination method to nd a general solution for the given linear
system
3.
x0 + 2 y
=
0
x0
=
0
y0
SOLUTION:
We can begin by eliminating x0 by subtrac
MATH 3310
Section 5.4 Solutions
Ryan Livingston
Section 5.4 Phase Planes
Page 272 1-13
In problems 1-2, veriy that the pair x (t) and y (t) is a solution to the given system. Sketch
the trajectory of the given solution in the phase plane
1.
dy
dx
= 3y 3 ;
MATH 3310 Differential Equations, Spring 2010 Instructor: Dr. Liancheng Wang Office: Sci. & Math. 511 E-mail: lwang5@kennesaw.edu Phone: 678-797-2139 Office Hours: 2:00-3:15 pm, 5:00-6:00 pm, T TH, or by appointment Text: A First Course of Differential Eq
Comparison of Dierentiation and integration Rules.
Let u be a dierentiable function of x. 1. d [x] = 1 dx dn [x ] = nxn1 , dx 1 d [ln |x|] = , dx x dx [e ] = ex , dx dx [a ] = (ln a)ax , dx d [sin x] = cos x, dx d [cos x] = sin x, dx d [tan x] = sec2 x, d
1) find the Laplace transforms - don't need to use the definition (no 0 to infinity. just use the formulas) 2) evaluate the inverse Laplace transforms 3) solve 2 de's Must use Laplace transforms
Differential Equations 1. determine whether or not the following functions are linearly independent or not 2. one solution given, use reduction formula (don't have to check if linearly independent) 4.3. homogeneous constant coefficient. degree two only (m
DE Exam 4 7.4 8.2 8.3 1) laplace transform 2) inverse laplace transform 3) solve homogeneous DE 2x2 distinct repeated complex 8.3 4) use the undetermined method 5) use the variation of parameters
Chapter 8. Systems of Linear First-Order Dierential Equations 8.3 Nonhomogeneous Linear Systems 1. Undetermined Coecients: X = AX + F. Ex. Solve
X = X =
6 1 6t X+ 43 10t + 4
1 2 8 X+ 1 1 3
2. variation of Parameters: Let (t) = [X1 , X2 ] be a fundame
Math 3310
Section 4.4 Solutions
Ryan Livingston
Section 4.4 Page 182: 9-31 odd
For Problems 9-25, Find the Particular Solution
9.
y + 2y y = 10
SOLUTION:
We begin by nding the homogeneous solution, so we dene y = ex so that the auxiliary equation is then
Math 3310
Section 4.3 Solutions
Ryan Livingston
Section 4.3 Page 173: 1-29
For Problems 1-19, Find the General Solution
1.
y + y = 0
SOLUTION:
We dene y = ex so the auxiliary equation is then
2 + 1 = 0
we solve for and get = i so the general solution is g
Math 3310
Section 4.2 Solutions
Ryan Livingston
Section 4.2 Page 166: 1-25 odd
In problems 1-12, nd a general solution to the dierential equation
1.
y 00 + 6y 0 + 9y = 0
SOLUTION:
We dene y = e
above to give
x
so that y 0 = e
x
2
and y 00 =
2
e
x
e
x
e x
MATH 3310 Exam 1 June 17, 2003 S. F. Ellermeyer Name Instructions. In order to receive full credit for any particular question, your solution must be mathematically correct and must be written correctly (using correct notation, etc.) Partial credit will b
MATH 3310 Final Exam
July 24, 2003
S. F. Ellermeyer
Name
Instructions. In order to receive full credit for any particular question, your solution
must be mathematically correct and must be written correctly (using correct notation, etc.)
Partial credit wi
Math 3310
Section 4.6 Solutions
Ryan Livingston
Section 4.6 Page 193: 1-11 odd
For Problems 1-8, Find the General Solution
1.
y + y = sec t
SOLUTION:
We nd the homogeneous solution by rst dening y = et so that the auxiliary equation becomes
2 + 1 = 0
so =
MATH 3310
Exercises on Higher Order ODEs
Exercises On Higher Order ODEs
For exercises 1-6, (a) nd the characteristic equation (b) the eigenvalues and (c) the corresponding eigenvectors for the matrix
Exercise 1
6 3
A=
2 1
SOLUTION:
(a) To nd the eigenvalu
Ryan Livingston
Section 1.3 Pg. 22 Exercises 1-7 odd
1.
(a) If we wanted to sketch a line that runs through the initial point (0, 2) that is in the direction
of the eld, we would get
If we recall that the ODE was
dy
= 2x + y
dx
with the initial condition
Ryan Livingston
Section 1.1 Pg. 5 Exercises 1-17
1.
d2 y
dx2
2x
dy
+ 2y = 0
dx
From the equation above, we can deduce that it is a second order ordinary dierential equation.
We can also deduce that the equation must be linear. Lastly, we can see that the
Ryan Livingston
Section 1.4 Eulers Method pg. 28 1-11 odd
1.
Use Eulers method to approximate the solution to the intital vlaue problem
dy
x
=
dx
y
y (0) = 4 at the points x = 0.1, 0.2, 0.3, 0.4 and 0.5 using steps of size 0.1 (h = 0.1)
Given the inital c
Ryan Livingston
Section 2.1 pg 46 1-25 odd
Determine wether or not the following equations are seperable
1.
dy
dx
sin (x + y ) = 0
SOLUTION:
The equation
dy
sin (x + y ) = 0
dx
is not seperable becasue we cannot break up the x and y variables in the term
Ryan Livingston
Section 2.3 pg 54 1-21 odd
Determine wether or not the following equations are seperable, linear, or both
1.
SOLUTION:
The
3.
SOLUTION:
The
5.
SOLUTION:
The equation can be put into standard form as
dr
3r = 3
d
so we can conlcude the it i
Ryan Livingston
2.5 Special Integrating Factors Page 67 1-14
In 1-6, Identify the equation as seperable, linear, exact, or having an integration factor that
is a function of either x alone of y alone.
1.
SOLUTION:
We can easily see that this equation is d
Chapter 2
Section 2.6
Ryan Livingston
2.6 Substitutions Page 79 9-27 odd
Use the method discussed under "homogeneous equations" to solve 9-16.
9.
SOLUTION:
The equation
xy + y 2 dx
can be placed into the form
dy
dx
x2 dy = 0
= f (x; y ) as
x2 dy = xy + y
Chapter 8. Systems of Linear First-Order Dierential Equations 8.2 Homogeneous Linear Systems 1. Homogeneous Linear Systems: X = AX where A is an n n matrix of constants. 2. Eigenvalues and Eigenvectors. Def: The polynomial equation det(A I ) = 0 is called