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 = @
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 =
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
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 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
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 23
4.3 The Method of Undetermined Coecients
In this section, we consider the nth order linear nonhomogeneous equation with constant
coecients:
a0 y (n) + a1 y (n1) + + an1 y + an y = g(t).
The general solution is
Example 1. Find the general
MA 266 Lecture 24
6.1 Denition of the Laplace Transform
Many practical engineering problems involve mechanical or electrical systems acted on by discontinuous or impulsive forcing terms. The methods introduced in Chapter 3 are often rather awkward
to use.
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 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 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
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)
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
Governing equation
mu + u + ku = F (t).
(1.1)
m is the m
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 equations
n-th order linear dierential equation in the gener
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. Find f (t) in
f (t) cos(t) = et sin(t).
3. Find y(t) in
y
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 constants.
Example 1. Find the solution v(t) to the following
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 the form
If M is a function of x only, and N is a functi
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 solve an rst order equation by integration on both sides.
Fall 2011 Exam 2 Solutions
DS
1 . Wephave x0 (t) = sin t, y 0 (t) = cos t, and z 0 (t) = 1. Make the substitution ds = x0 (t)2 + y 0 (t)2 + z 0 (t)2 dt to obtain
Z
Z
2
sin t sin t
y sin(z)ds =
p
sin2 t + cos2 t + 1 dt =
Z
2
0
C
2
sin2 t dt
0
From the id