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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
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
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
MATH 3310 Differential Equations, Spring 2010 Instructor: Dr. Liancheng Wang Office: Sci. & Math. 511 E-mail: [email protected] Phone: 678-797-2139 Office Hours: 2:00-3:15 pm, 5:00-6:00 pm, T TH, or
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]
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.
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. var
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 Eigenvector
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, s
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
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 =
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 correct
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 cor
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 = e
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
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
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 th
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
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 becasu
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 st
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.
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
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
Chapter 7. The laplace Transform 7.3 Operational Properties I 1. Translation on the s-axis: First Translation Theorem: If F (s) = Lcfw_f (t) and a is any real number, then Lcfw_eat f (t) = F (s a). Ex