LECTURE 5: FIRST ORDER LINEAR DIFFERENTIAL EQUATION AND THE EXACT
EQUATIONS
MINGFENG ZHAO
September 18, 2015
First order linear dierential equations and the integrating factors
Lets solve y + p(x)y = f (x). Multiply r(x) on the both sides, we get r(x)y +
LECTURE 20: THE LAPLACE TRANSFORMS OF INTEGRAL, CONVOLUTION AND DIRAC
DELTA
MINGFENG ZHAO
October 28, 2015
The Laplace transform
Denition 1. Let f (t) be a function on [0, ), then
I. The Laplace transform of f , denoted by L[f ](s), is dened as:
f (t)est
LECTURE 18: THE LAPLACE TRANSFORM AND TRANSFORMS OF DERIVATIVES AND
ODES
MINGFENG ZHAO
October 23, 2015
Example 1. A mass of 4 kg on a spring with k = 4 and a damping constant c = 1. Suppose F0 = 2. Using forcing
function F0 cos(t), nd the that causes pra
LECTURE 21: THE LAPLACE TRANSFORMS OF INTEGRAL, CONVOLUTION AND DIRAC
DELTA
MINGFENG ZHAO
October 30, 2015
The Laplace transform
Denition 1. Let f (t) be a function on [0, ), then
I. The Laplace transform of f , denoted by L[f ](s), is dened as:
f (t)est
LECTURE 13: MECHANICAL VIBRATIONS
MINGFENG ZHAO
October 07, 2015
Mass-Spring System:
Figure 1. Mass-Spring System
Let x(t) be the displacement of the mass, then
mx + cx + kx = F (t) .
For free motion (that is, F (t) = 0), rewrite the equation, we have
2
x
LECTURE 1: INTRODUCTION TO THE DIFFERENTIAL EQUATIONS
MINGFENG ZHAO
September 09, 2015
Introduction to the dierential equations
Denition 1. A dierential equation is an equation involving the derivatives of the dependent variable.
If the
highest order of t
LECTURE 6: EXACT EQUATIONS AND INTEGRATE FACTORS
MINGFENG ZHAO
September 21, 2015
Exact equations
Denition 1. Consider the dierential equation M (x, y)+N (x, y)y = 0, we say that the dierential equation M (x, y)+
N (x, y)y = 0 is exact if My (x, y) = Nx (
LECTURE 2: SEPARABLE DIFFERENTIAL EQUATIONS
MINGFENG ZHAO
September 11, 2015
Example 1. Verify that x(t) = Ce2t is a solution to x = 2x. Find C to solve for the initial condition x(0) = 100.
In fact,
x = 2Ce2t = x = 2x = x(t) = Ce2t is a solution to x = 2
LECTURE 4: FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS AND THE
INTEGRATING FACTORS
MINGFENG ZHAO
September 16, 2015
Theorem of the existence and uniqueness for the rst order dierential equations with
the initial value condition
Theorem 1 (Picards Theorem on
LECTURE 6: EXACT EQUATIONS AND INTEGRATE FACTORS, AND AUTONOMOUS
EQUATIONS
MINGFENG ZHAO
September 23, 2015
To solve an exact equation
Let M (x, y) + N (x, y)y = 0 be an exact equation, that is, My (x, y) = Nx (x, y), then we can nd (x, y) such that
x (x
LECTURE 6: AUTONOMOUS EQUATIONS AND EULERS METHOD
MINGFENG ZHAO
September 25, 2015
Phase diagram of the autonomous equation
An autonomous equation is of the form y = f (y), if f (a) = 0, y(x) a is called an equilibrium solution, and a is
called a critical
LECTURE 10: HOMOGENEOUS SECOND ORDER LINEAR ODES WITH CONSTANT
COEFFICIENTS
MINGFENG ZHAO
September 30, 2015
Theorem 1. Let p(x) and q(x) be continuous functions, y1 and y2 are two linearly independent solutions to a homogeneous equation y + p(x)y + q(x)y
LECTURE 19: THE LAPLACE TRANSFORM AND TRANSFORMS OF DERIVATIVES AND
ODES
MINGFENG ZHAO
October 26, 2015
The Laplace transform
Denition 1. Let f (t) be a function on [0, ), then
I. The Laplace transform of f , denoted by L[f ](s), is dened as:
f (t)est dt,
LECTURE 23:THE LAPLACE TRANSFORM AND SYSTEMS OF ODES
MINGFENG ZHAO
November 02, 2015
The Laplace transform
Denition 1. Let f (t) be a function on [0, ), then
I. The Laplace transform of f , denoted by L[f ](s), is dened as:
L[f (t)](s) =
f (t)est dt, fo
LECTURE 16: NONHOMOGENEOUS EQUATIONS AND FORCED OSCILLATIONS AND
RESONANCE
MINGFENG ZHAO
October 19, 2015
Theorem 1. Let yc (x) be the general solution to the homogeneous equation y + p(x)y + q(x)y = 0 ( which is call the
complementary solution), and yp (
LECTURE 4: FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS AND THE
INTEGRATING FACTORS
MINGFENG ZHAO
September 16, 2015
Theorem of the existence and uniqueness for the rst order dierential equations with
the initial value condition
Theorem 1 (Picards Theorem on
LECTURE 6: EXACT EQUATIONS AND INTEGRATE FACTORS
MINGFENG ZHAO
September 21, 2015
Exact equations
Denition 1. Consider the dierential equation M (x, y)+N (x, y)y = 0, we say that the dierential equation M (x, y)+
N (x, y)y = 0 is exact if My (x, y) = Nx (
LECTURE 8: AUTONOMOUS EQUATIONS AND EULERS METHOD
MINGFENG ZHAO
September 25, 2015
Phase diagram of the autonomous equation
An autonomous equation is of the form y = f (y), if f (a) = 0, y(x) a is called an equilibrium solution, and a is
called a critical
LECTURE 7: EXACT EQUATIONS AND INTEGRATE FACTORS, AND AUTONOMOUS
EQUATIONS
MINGFENG ZHAO
September 23, 2015
To solve an exact equation
Let M (x, y) + N (x, y)y = 0 be an exact equation, that is, My (x, y) = Nx (x, y), then we can nd (x, y) such that
x (x
LECTURE 9: EULERS METHOD, AND SECOND ORDER LINEAR ODES
MINGFENG ZHAO
September 28, 2015
Numerical methods: Eulers method
Consider the initial value problem:
y = f (x, y),
y(x0 ) = y0 .
Recall Taylor expansion:
y(x + x) = y(x) + y (x)x +
y (x)
(x)2 + .
2
L
LECTURE 10: HOMOGENEOUS SECOND ORDER LINEAR ODES WITH CONSTANT
COEFFICIENTS
MINGFENG ZHAO
September 30, 2015
Theorem 1. Let p(x) and q(x) be continuous functions, y1 and y2 are two linearly independent solutions to a homogeneous equation y + p(x)y + q(x)y
LECTURE 3: SLOPE FIELDS
MINGFENG ZHAO
September 14, 2015
Separable dierential equations
For a separable dierential equation:
y =
dy
= f (x)g(y).
dx
There are two cases:
Case I: If g(a) = 0 for some constant a, then y(x) a is a solution.
1
dy
= f (x)g(y),
LECTURE 11: HOMOGENEOUS SECOND ORDER LINEAR ODES WITH CONSTANT
COEFFICIENTS AND MECHANICAL VIBRATIONS
MINGFENG ZHAO
October 02, 2015
Theorem 1. Let p(x) and q(x) be continuous functions, y1 and y2 are two linearly independent solutions to a homogeneous eq
LECTURE 14: NONHOMOGENEOUS EQUATIONS
MINGFENG ZHAO
October 14, 2015
Theorem 1. Let yc (x) be the general solution to the homogeneous equation y + p(x)y + q(x)y = 0 ( which is call the
complementary solution), and yp (x) be any particular solution to the n
LECTURE 15: NONHOMOGENEOUS EQUATIONS
MINGFENG ZHAO
October 16, 2015
Theorem 1. Let yc (x) be the general solution to the homogeneous equation y + p(x)y + q(x)y = 0 ( which is call the
complementary solution), and yp (x) be any particular solution to the n
LECTURE 12: MECHANICAL VIBRATIONS
MINGFENG ZHAO
October 05, 2015
RLC Circuit System
There is a resistor with a resistance of R ohms, an inductor with an inductance of L henries, and a capacitor with
capacitance of C farads. There is also an electric sourc
LECTURE 17: FORCED OSCILLATIONS AND RESONANCE
MINGFENG ZHAO
October 21, 2015
Mass-Spring System:
Figure 1. Mass-Spring System
Let x(t) be the displacement of the mass, then
mx + cx + kx = F (t) .
We are interested in periodic forcing, F (t) = F0 cos(t) wi
Math 215/255 Final Exam (Dec 2005)
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MIDTERM EXAM TOPICS: MATH 215/255 FEBRUARY 2016
Note: Any topic from any part of the course could be tested either individually or in combination with
other course topics.
First Order Differential Equations
Identify linear, nonlinear and separable equati
Math 392 Exam 1 Fall 2010
Name: ML
Instructions:
1. Make sure you have all 10 pages of the test (including this cover page).
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Correct answers Without s
EUS Review Problem Set Solutions
Mathematics 255 - Midterm 2
Ordinary Differential Equations
Note on notation: Whenever log (x) is used without a subscript to indicate the base, it is assumed to be
base e in math courses. Thus in this review package, log
s
er
w
ns
la
na
Fi
ti
rip
sc
n
Final Answers
MATH215 December 2013
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