MA 266 Project 2
RLC Circuits
3/22/2014
1. Setting up the solution:
Function for plotting solution for a given value of
:
function PlotForW(func, tRange, xInitial, W)
%plotForE Plot differential equa
Shu Yi Neoh
0026439252
Computer Project 1. Nonlinear springs
1. Eps = 0.0
Ma266-Project 1
Prof. Joseph Chen 1
Shu Yi Neoh
0026439252
Eps=0.2
Ma266-Project 1
Pro
Hemanth Mullangi Chenchu
MA 266
Project 3
1
PROJECT 3: Predator-Prey Equations
Question 1:
Graph 1: Plot of the trajectory through the point (0.8, 0.4).
From the graph, we can observe that as t increa
Shrish Mansey
Predator
Prey
Equations
0025775359
MA 266
Project 3
Shrish Mansey
0025775359
Project 3
Predator Prey Equations
Problem 1:
Assuming that there are 800,000 aphids and 400,000 ladybugs, as
PROJECT2:RLCCircuits
SanchariniChakraborty
Date:23/07/2014
Question 1:
If Q (t) = charge on a capacitor at time t in an RLC circuit and E (t) = applied voltage, then Kirchhoffs Laws
gives the followin
1. Let E = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 (hard spring) and plot the solutions of the above initial value
problem for 0 t 20.
a. Estimate the amplitude of the spring.
The amplitude approaches _ as t app
Ordinary Dierential Equation: Chapter 3.1
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/6
Second order ODE
Second Order ODE
Second Order ODE
A second order ODE is usua
Ordinary Dierential Equation: Chapter 2.6
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/8
Exact ODE
Exact ODE
Exact ODE
Integrating factor revisited: example
dy
2
+ y
Ordinary Dierential Equation: Chapter 2.7
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/3
Eulers method
Eulers method
Eulers method
For a general rst order ODE
dy
= f
Ordinary Dierential Equation: Chapter 2.4
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/8
Nonlinear ODE and linear ODE
Nonlinear v.s. linear
Linear and nonlinear ODE
O
Ordinary Dierential Equation: Chapter 2.5
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/5
Autonomous ODE
Autonomous ODE
Autonomous ODE and Equilibrium
Autonomous ODE i
Ordinary Dierential Equation: Chapter 3.2
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/7
Second order ODE
Second Order ODE
Notation of dierential operator
Let p(t), q
Running head: VARYING DEFINITIONS OF ONLINE COMMUNICATION
1
Green text boxes
contain explanations
of APA style
guidelines.
The title
should
summarize
the papers
main idea and
identify the
variables
un
Ordinary Dierential Equation: Chapter 3.3
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/7
Second order ODE
Linear Second Order ODE with constant coecient
Linear Second
Ordinary Dierential Equation: Chapter 2.1
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/4
First Order Linear Ordinary Dierential Equation
First Order linear ODE
First
MA 266 Lecture 4
Section 2.2
Separable Equations
Note: we use x (instead of t) as the independent variable in this section.
The general form of a nonlinear rst order equation is
It can be written in t
MA 266 Lecture 2
Section 1.2
Solutions of Some Dierential Equations
In this section, we discuss how to solve the dierential equation of the following form
dy
= ay b,
dt
where a and b are given constan
MA 266 Lecture 1
Section 1.1
Mathematical Models; Direction Fields
Question: What is a dierential equation?
A dierential equation is
Example 1. (Types of equations)
1. Find x in
x2 + 2x + 1 = 0.
2. Fi
Ordinary Dierential Equation: Chapter 4.1&4.2
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/7
Higher order linear ODE
Higher order linear ODE
Higher order linear equat
Ordinary Dierential Equation: Chapter 3.8
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/5
Vibrating system
Forced vibration
Spring motion with specied external force
G
Ordinary Dierential Equation: Chapter 3.7
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1 / 11
Vibrating system
Vibrating system
Spring system: part 1
We study the appli
Ordinary Dierential Equation: Chapter 3.5
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/1
Nonhomogeneous ODE
Nonhomogeneous ODE
y + p(t)y + q(t)y = g (t)
where p(t), q
Ordinary Dierential Equation: Chapter 3.4
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/7
Repeated Roots and Reduction of Order
Repeated roots
Repeated roots
Homogeneo
Ordinary Dierential Equation: Chapter 3.6
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/6
Variation of constant
Variation of constant
Variation of constant: part 1
Sup
MA 266 Lecture 16
Section 3.4
Repeated Roots; Reduction of Order
Review. Consider the linear homogeneous equation with constant coecient
ay + by + cy = 0.
The characteristic equation is
If the discrim
MA 266 Lecture 3
Section 2.1
Factors
Linear Equations; Method of Integrating
In this section, we consider the rst order linear equation.
The general form is
The standard form is
Sometimes we can sol
Ordinary Dierential Equation: Chapter 2.2
Yongyong Cai
Department of Mathematics
Purdue University, West Lafayette, IN
1/5
Modelling with First Order Dierential Equation
Modelling with First Order Die
Equations :
Q + 4Q + 5Q = 10cos(wt), where Q is function of time.
Initial conditions:
Q(0) = Q(0) = 0
Using oed45 in mathlab the following plot were obtained.
=0
2.5
2
1.5
1
0.5
0
0
10
20
30
40
50
60
y f ( x) , y
y , y
6.1
6.2
6.3
6.4
Matlab
6.1
1 (1,2),
M ( x,
y)
2x
: y = y(x) ,
:
dy
2 x
dx
y x 1 2
2
y 2x d x x C
(C )
C = 1, y x 2 1 .
2 m
s
t s(t)
s(t)
d 2s
m 2 mg
dt
,
d
2
x2 2x3 = 0 x = 3 1
(a) y 3x 1
(b)
d2y
dx 2
x2 2 x 1
(c) xy 2 y y x 1
(d) y y 0
(e) y ky
10-1
y = f(x)
3
C y = x2 + C y' = 2x
: y' = 2x y = x2 + C y' = 2