Math 207
Homework 2
Due on Monday, October 17, 2011 at 11:30 a.m.
Read Carefully: Write your name at the top of this sheet of paper and staple it to the front of your
homework assignment. Your work must be organized, clear, and mathematically precise. All
Chapter 8 Systems of Linear First-Order
Differential Equations
dk y( t)
(1) Section 4.8: k
dt
dk y( t)
(2) Chapter 7:
dt k
Dk y ( t )
s kY ( s ) s k 1 y ( 0 ) s k 2 y ( 0 ) LL y ( k 1) ( 0 )
(3) Chapter 8: Using matrix operations
1
(1)
3
linear & consta
4-4 Undetermined Coefficients
Superposition Approach
This section introduces some method of guessing the particular
solution.
4-4-1 ?
(1)
(2)
Suitable for linear and constant coefficient DE.
an y ( n ) ( x ) +an- 1 y ( n- 1) ( x ) +L +a1 y( x ) +a0 y =g
?, ?
?
1
Chapter 6 Series Solutions of Linear Equations
? DE ? solutions ? polynomial ?
? Cauchy-Euler Method ? Taylor Series ?
y ( x ) = cn ( x - x0 )
n
n=
0
? power series centered at x0
x0 is a non-singular point (Sec. 6-1)
regular singular point (Sec.
Chapter 5 Modeling with Higher Order
Differential Equations
Chapter 4 ?
?, ? linear DE ?
? linear DE with constant coefficients
1
5-1 Linear Models: Initial Value Problem
?: x,
d x
?: ?
dt
F =ma
F - b v =ma
d2 x
dt 2
d 2 x =F
m 2
dt
d 2 x +b dx =F
m 2
dt
Methods of Solving the First Order Differential Equation
graphic method
numerical method
analytic method
direct integration
separable variable
method for linear equation
method for exact equation
homogeneous equation method
Bernoullis equation method
ser
Chapter 4 Higher Order Differential Equations
Highest differentiation:
dny
n , n > 1
dx
Most of the methods in Chapter 4 are applied for the linear DE.
1
? DE ?
Homogeneous
Constant coefficients
Linear
Cauchy-Euler
DE
Nonhomogeneous
Homogeneous
Nonhomogen
2-4 Exact Equations
2-4-1
first order DE
M ( x, y ) dx + N ( x, y ) dy = 0
M ( x, y ) = N ( x, y )
(1) y
x
(2)
Exact Equation
M ( x, y ) N ( x , y )
y
x
is independent of x
M ( x, y )
M ( x, y ) N ( x , y )
y
x
N ( x, y )
is independent of y
Modified
Chapter 7 The Laplace Transform
?: ?
dk y(t)
Chapter 4 ?
dt k
Laplace transform
Dk y ( t )
d k y ( t ) ?
dt k
s k Y ( s ) - s k - 1 y ( 0 ) - s k - 2 y( 0 ) - LL - sy ( k - 2) ( 0 ) - y ( k - 1) ( 0 )
1
Section 7-1 Definition of the Laplace Transform
7-1-