MATH-20D
Introduction to Differential Equations
Welcome to Lecture B
Instructor: James Hall
TA: William Woods
Lecture - Monday, Wednesday, Friday 2-2:50PM
Tu 4, 5, 6, 7 PM
TA: Francesca Grogan
Tu 12, 1, 2, 3 PM
Course Material
Textbook: Elementary Differe
Fact Sheet for 20D Exam 1
1. First Order Equations
(Integrating Factors) The general rst order linear equation is of the from:
(1)
y + p(t)y = g(t) .
One can always nd the general solution to this equation by rst nding
P (t) so that:
P
= p,
and then comp
2015/9/25
Math20D:OrdinaryDifferentialEquations
MATH 20D Fall 2015
Introduction to Differential Equations
Instructor: Jacob Sterbenz
Links
20D Home Syllabus Meetings Schedule Homework
MATLAB Exams Help
Lecture Schedule
Week
Monday
Wednesday
Friday
1
2
3
S
2015/9/25
Math20D:OrdinaryDifferentialEquations
MATH 20D Fall 2015
Introduction to Differential Equations
Instructor: Jacob Sterbenz
Links
20D Home Syllabus Meetings Schedule Homework
MATLAB Exams Help
Basic Syllabus
Course: Math 20D
Title: Introduction t
Yesha Luo
A92417246
A02
Gautam Wilkins
Matlab Assignment 1
1. Input: > log(20)/log(8)
Output: ans =
1.4406
2. The previous command computes a increasing range of a series number from -3 to 12, which
adds 0.4 in turn. The last value is 11.8 not 12 is that
4.5 DIMENSION OF A VECTOR SPACE
DIMENSION OF A VECTOR SPACE
Define the size of a vector space by the size of basis.
Def: If basis of V is a finite set, then V is said to be
finite-dimensional, and the dimension of V, dim V,
is the number of vectors in t
4.6 Rank
If A Rmn, each row of A is a vector in Rn.
1 2 3
3
A
, and R1 is 1, 2,3 .
4 5 6 23
Def: The subspace spanned by the row vectors is
the row space of A (Row A).
Row A is a subspace of Rn.
Since the rows of A are identical to the columns of
AT, Ro
Direction Fields
Studying Solutions Without Finding Them (Explicitly)
First Things First
Whats A Differential
Equation?
Any equation that involves a
derivative is a differential equation
0
y = (1
t) y
Derivatives Tell Us Change
0
y = (1
Rate of
Change
of
First Order Linear
Equations
Integrating Factors
First Order Linear Equations
Only First Derivative
a 0 ( t) y 0 + a 1 ( t) y = g ( t)
Of the form:
Notice, we can rewrite:
a 1 ( t)
g ( t)
y+
y=
a 0 ( t)
a 0 ( t)
0
First Order Linear Equations
Only First D
First Order Linear
Equations
Integrating Factors
First Order Linear Equations
Only First
Derivative
Of the form:
Notice, we can rewrite:
First Order Linear Equations
Only First
Derivative
Of the form:
Notice, we can rewrite:
So we consider general form:
R
Direction Fields
Studying Solutions Without Finding Them (Explicitly)
First Things First
Whats A Differential
Equation?
Any equation that involves a
derivative is a differential equation
Derivatives Tell Us Change
Rate of
Change
of y
Current
time
Value of
Separation of
Variables
Solving First Order Differential Equations
Solving ODEs
What is Solving an ODE?
Eliminating All Derivatives
y = g ( t)
Explicit Form
f ( y ) = g ( t)
Implicit Form
This Chapter
1st Order (Only First Derivative)
Linear and Nonlinear
MATH-20D
Introduction to Differential Equations
Welcome to Lecture B
Instructor: James Hall
TA: William Woods
Lecture - Monday, Wednesday, Friday 2-2:50PM
Tu 4, 5, 6, 7 PM
TA: Francesca Grogan
Tu 12, 1, 2, 3 PM
Course Material
Textbook: Elementary Differe
Separation of
Variables
Solving First Order Differential Equations
Solving ODEs
What is Solving an ODE?
Eliminating All Derivatives
Explicit Form
Implicit Form
This Chapter
1st Order (Only First Derivative)
Linear and Nonlinear
Calculus Brain Teaser:
?
Ca
Math 20A. Lecture 3.
Linear functions and constant rates of change
A function is called linear if its graph is a line. Rates of change of these functions can be found without
calculus.
Example 1 (a) A van is s = 200 + 60t miles east of a truck stop at tim