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 equation solution solved with Runga Kutta Method
xPrime = @
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 increases, the ladybug population and the aphid
population in
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 t increases, the
aphid and the ladybug population fluct
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 approaches 20
b. What appears to happen to the amplitude a
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 following second order differential equation for Q (t):
If L =
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(t) be continuous functions on an interval
I = (, ), wh
MA 266 Lecture 8
Section 2.4
Equations
Dierences Between Linear and Nonlinear
Does every initial value problem have exactly one solution?
For a linear equation y + p(t)y = g(t), we have the following fundamental theorem
Theorem (linear equation) If the fu
MA 266 Lecture 12
Section 2.7 Numerical Approximation: Eulers Method
In this section, we demonstrate how to implement Eulers method with MATLAB.
Download the Matlab code for Eulers method at http:/math.rice.edu/deld/
(search Euler on the webpage)
Review:
MA 266 Lecture 10
Section 2.6
Exact Equations and Integrating Factors
In the section, we consider a special class of rst order equations known as exact equations.
Example 1. Solve the dierential equation
2x + y 2 + 2xyy = 0.
Consider the dierential equati
MA 266 Lecture 13
Section 3.1
Coecients
Homogeneous Equations with Constant
Terminologies
Starting from this section, we study second order dierential equation of the form
The above equation is called linear if
For a linear equation, we usually write it a
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 usually written as
y = f (t, y , y )
for some function f
It
MA 266 Lecture 15
Section 3.3
Complex Roots of Characteristic Equation
Review. Consider the linear homogeneous equation with constant coecient
ay + by + cy = 0.
The characteristic equation is
If the discriminant b2 4ac > 0, then
In this section we study t
MA 266 Lecture 18
Section 3.6
Variation of Parameters
In this section, we introduce a more generally applicable approach to nd particular solution
of nonhomogeneous equation. The method is called variation of parameters.
We will use the following example
MA 266 Lecture 14
Section 3.2 Solutions of Linear Homogeneous Equations; Wronskian
Terminologies
In this section, we study the structure of solutions of second order linear dierential equation.
Let p and q be continuous functions on an open interval I. Fo
MA 266 Lecture 19
Section 3.7
Mechanic and Electrical Vibrations
In this section we use second order linear equations to model the motion of a mass on a spring.
Consider a mass m hanging at rest on the end of a vertical spring of original length l.
The ma
MA 266 Lecture 17
Section 3.5
Nonhomogeneous Equations; Method of
Undetermined Coecients
We consider the nonhomogeneous equation
y + p(t)y + q(t)y = g(t).
(1)
The corresponding homogeneous equation is
Structure of solution
Theorem The general solution of
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 discriminant b2 4ac > 0, then
If the discriminant b2 4ac < 0,
MA 266 Lecture 20
Section 3.7 Mechanic and Electrical Vibrations (contd)
Review For undamped free vibrations, the governing equation is
Example 1. (Problem 6) A mass of 100 g stretches a spring 5cm. If the mass is set in
motion from its equilibrium positi
MA 266 Lecture 21
Section 3.8
Forced Vibration
In this section, we consider the situation in which a periodic external force is applied to a
spring-mass system.
Forced Vibration with Damping
Example 1. Suppose that the motion of a certain spring-mass syst
MA 266 Lecture 22
4.1 General Theory of nth Order Linear Equations
An nth order linear dierential equation is
If P0 is nowhere zero in the interval I, the equation can be written as
The initial conditions are
Theorem (Existence and Uniqueness) If p1 , p2
MA 266 Lecture 11
Section 2.7 Numerical Approximation: Eulers Method
In this section, we introduce a numerical method for solving the rst order initial value
problem
dy
= f (t, y),
y(t0 ) = y0 .
dt
The method is called
or
How to use tangent lines to appro
MA 266 Lecture 9
Section 2.5
Dynamics
Autonomous Equations and Population
A dierential equation is called
Autonomous equations are
if it has the form
.
In this section, we will use geometrical methods to obtain important qualitative information about dier
MA 266 Lecture 5
Section 2.2
Separable Equations (contd)
Example 1. Consider the initial value problem
y =
ty(4 y)
,
3
y(0) = y0 > 0.
(a).Determine how the behavior of the solution as t increases depends on the initial value y0 .
(b).Suppose that y0 = 0.5
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 Order ODE with constant coecient
Homogeneous ODE with
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
Suppose we want to solve the nonhomogeneous ODE
y + p(t)y
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
Homogeneous ODE with constant coecients,
ay + by + cy = 0
Charac
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(t), g (t) functions dened on interval I
Associated hom
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 applications of second order ODE
ay + by + cy = g (t),
y (0)
MA 266 Lecture 6
Section 2.3
Modeling with First Order Equations
In this section, we consider mathematical models using rst order dierential equations.
Steps for mathematical modeling:
1. Construction of the Model
2. Analysis of the Model
3. Comparison wi