LECTURE 4, LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS
PO-NING CHEN
Today we will learn the classification of first order equation and learn how to solve the
inear first order equation
y 0 + p(x)y = q(x)
using the method of integrating factors.
(1) Classifi
Lecture 3
Linear equation
Topics for today
Today, we will learn the following topics:
(1) Classification of first order equations.
(2) Linear first order equations
Classification of first order equations
We also observed that a differential equation might
Lecture 15
Variation of Parameter
Mar 7th, 2017
Outline
Today, we use the method of Variation of Parameter to solve
y 00 + p1 (x)y 0 + p2 (x)y = q(x)
where p1 (x), p2 (x) and q(x) can be arbitrary functions.
The method is based on the solution of the homo
LECTURE 4, LINEAR EQUATION AND BERNOULLI EQUATION
PO-NING CHEN
Today we derive the formula for the solution of
y 0 + p(x)y = q(x)
(1)
(2)
(3)
(4)
The general formula
Some examples
Initial value problems
Bernoulli equation
1. The general formula
Consider t
Lecture 7
Exact equation
Jan 31, 2017
Exact equation
Today, we begin our discussion on exact equation. Here are the topics for
today
1
Partial Derivatives
2
Implicit differentiation using partial derivatives
3
Exact equation and its solution
4
Verifying w
LECTURE 10: 2ND ORDER HOMOGENOUS EQUATION WITH
CONSTANT COEFFICIENT.
PO-NING CHEN
Today, we will learn to solve 2nd order homogenous equation with constant coefficient
y 00 + ay 0 + by = 0
using the characteristic equation.
1. Review of 2nd order homogeno
Lecture 4
Linear equation and
Bernoulli equation.
Topics for today
Today, we will cover the following topics:
(1) Formula for solution of linear equations.
(2) Initial Value Problems for linear equation
(3) Bernoulli equation
gGeneral formula
We also obs
Lecture 12
Complex root
Feb 23, 2017
Outline
In this lecture, we will solve
y 00 + ay 0 + by = 0
assuming
r 2 + ar + b = 0,
has complex roots.
Lecture 12 Complex root
Feb 23, 2017
2 / 17
Polynomials with no root
There are many 2nd order polynomial
r 2 + a
LECTURE 5: SEPARABLE EQUATION
PO-NING CHEN
Today we will first review the Bernoulli equation and then we will derive the general
solutions of the separable equations.
y0 =
(1)
(2)
(3)
(4)
(5)
F (x)
G(y)
Review of Bernoulli equation
General Solution of sep
LECTURE 2, WHAT IS DIFFERENTIAL EQUATION?
PO-NING CHEN
Here are the things we will learn today:
(1)
(2)
(3)
(4)
(5)
(6)
What is a differential equation?
What is an ordinary differential equation?
What is the order of a differential equation?
How to verify
Math 46 Sec 1
Ordinary Differential Equation
What is a differential Equation?
We learned a lot about solving equations in pre-calculus
and calculus. For example, we learned to solve
x^2+ 3x+2=0
or
sin(y) +y =0
where the unknowns in the equations are numbe
Lecture 16
Higher order linear equation
Mar 8, 2016
Outline
Today, we introduce higher order linear equation.
We will first go over the general theory of higher order linear equation and
homogenous equation.
Then, we study higher order homogenous equation
Lecture 11
Repeated root and reduction of order
Feb 21, 2017
Outline
Last time, we studied 2nd order linear equation with constant coefficients
y 00 + ay 0 + by = 0
where the characteristic equation
r 2 + ar + b = 0
has a repeated root.
This is also a nat
Lecture 17
Homogenous equation with constant coefficient
Mar 14, 2017
Outline
In this lecture, we look at homogenous equation with with constant
coefficients.
Namely, we study equations of the following form
y (n) + an1 y (n1) + + a1 y (1) + a0 y = 0.
Lec
Lecture 6
Homogeneous equation.
Topics for today
Today, we will solve the homogenous equation
y=f(y/x)
(1) Some examples
(2) General Solution and more examples
General Solution
We also observed that a differential equation might have many solutions.
To s
Lecture 9
Integrating factor for non-exact equation
Feb 14, 2017
Exact equation
We will look at what we can do if the differential equation is not exact.
In the second half of the class, we will begin the discussion on second
order differential equation.
Lecture 8
Exact equation II
Feb 2, 2017
Exact equation
Today, we will finish our discuss on exact equation.
We will first see how it relates to what we learned about solving linear
equation and separable equation in the last two weeks.
Lecture 8 Exact equ
Lecture 2
What is a differential
equation?
Topics for today
Today, we will learn the following topics:
(1) What is a differential equation?
(2) What is a solution?
(3) Particular solution and general solution.
(4) Initial Value Problems
What is a differe
Lecture 14
Method of undetermined coefficients
Mar 2, 2017
Outline
Today, we use the method of undetermined coefficients to solve
y 00 + ay 0 + by = q(x)
where q(x) is either a exponential function, polynomial or sine or cosine
function.
Lecture 14 Method
Lecture 5
Separable equation.
Topics for today
Today, we will solve the separable equation
y= F(x)/ G(y)
(1) General Solution.
(2) An example related to linear equation
(3) More examples and initial value problem
First, we will review the Bernoulli equat