So far, we have looked at homogeneous equations
|
|
- Florence Goodwin
- 8 years ago
- Views:
Transcription
1 Chapter 3.6: equations Non-homogeneous So far, we have looked at homogeneous equations L[y] = y + p(t)y + q(t)y = 0. Homogeneous means that the right side is zero. Linear homogeneous equations satisfy the superposition principle: sums of solutions are solutions. We now look at non-homogeneous equations: L[y] = y + p(t)y + q(t)y = g(t), where the RHS is not necessarily zero. 1
2 The New Issue It is: Find a solution of an inhomogeneous equation, say Y (t): L[Y ] = Y + p(t)y + q(t)y = g(t). The general solution is then: y(t) = Y (t) + c 1 y 1 (t) + c 2 y 2 (t) where y 1, y 2 are a fundamental set of solutions of the homogeneous equation L[y] = 0. Reason: L[y] = L[Y ] + L[c 1 y 1 + c 2 y 2 ] = L[Y ] + 0 = g. 2
3 General solution Let s make sure that y(t) = Y (t) + c 1 y 1 (t) + c 2 y 2 (t) is the general solution of L[y] = g. you have another, say y other. Then Suppose L[y] = g = L[y other ] = L[y y other ] = 0 = y y other = c 1 y 1 + c 2 y 2, for some c 1, c 2, since y y other solves the homogeneous equation, and the RHS is the general solution. 3
4 How to solve L[Y ] = g The first method is the method of undetermined coefficients = MUC. It only applies when you know in advance what kind of solution the equation will have. In the next chapter, a more systematic method, variation of parameters, will be used. But for special equations such as constant coefficient equations, where the RHS g is an exponential function, MUC is quicker. This motivates learning how to guess the type of solution. 4
5 How to solve L[Y ] = g for CC equations Example: Find a solution of y 3y 4y = 3e 2t. Key fact (which we already know): if Ly = ay + by + cy, then L[e rt ] = (ar 2 + br + c)e rt. Thus, in our problem, L[e rt ] = (r 2 3r 4)e rt. When solving the homogeneous equation, we wanted the RHS = 0 so we chose r to be a solution of the characteristic equation. But now, we want the RHS to be 3e 2t. Clearly, we have to pick r = 2. 5
6 Choosing the multiple We can t always solve with Y = e 2t but we can try Y = Ae 2t and solve for A. Indeed, L[Ae 2t ] = A( )e 2t = 3e 2t 6A = 3 A = 1 2. Thus, one solution is Y = 1 2 e2t. Since it is only one solution, we call it a particular solution. To find the general solution, we need to add the general solution of the homogeneous problem. (We won t do it, because that was last chapter). 6
7 Harder example Find a solution of: y 3y 4y = 2 sin t. This is harder because d dt sin t = cos t, d dt cos t = sin t. So sines and cosines are not quite preserved by the derivative. So we have to try: Y = A sin t + B cos t. Functions of this kind are preserved by taking a derivative. 7
8 Choosing the coefficients Write: L[y] = y 3y 4y. We claim that there exist constants A, B so that L[A sin t + B cos t] = 2 sin t. A bit of computation shows: L[A sin t + B cos t] = A( sin t 3 cos t 4 sin t) + B( cos t + 3 sin t 4 cos t). So we need: A( sin t 3 cos t 4 sin t) +B( cos t + 3 sin t 4 cos t) = 2 sin t. 8
9 Choosing the coefficients Equivalently, ( A 4A + 3B) sin t +( 3A B 4B) cos t = 2 sin t 5A + 3B = 2, 3A 5B = 0. Thus, A = 5 3 B = 3B B = 2 B = So A = 5 17 and Y = 5 17 sin t cos t. 9
10 A disease Unfortunately, this method does not work if the RHS g is a solution of the homogeneous equation L[g] = 0 on the LHS. For instance, y + y = sin t. You cannot just try C 1 cos t + C 2 sin t since the LHS will kill it. The cure is, as in reduction of order, to multiply by t. Try y = C 1 t cos t + C 2 t sin t. Only do this when the RHS is a homogeneous solution! If you do this, you will find that terms with a t in front are killed. So what remains is the equation 2C 1 sin t + 2C 2 cos t = sin t. The factor of 2 comes from (yt) = 2y + ty. 10
11 Example Solve the initial value problem: y + 4y = sin 2t, y(0) = 0, y (0) = 1. The RHS is a solution of the homogeneous equation y +4y = 0 so we need to try At cos 2t+ Bt sin 2t. The equation becomes: 4A sin 2t+4B cos 2t = sin 2t = A = 1 4, B = 0. The general solution is: y = C 1 cos 2t + C 2 sin 2t 1 t cos 2t. 4 Then, y(0) = C 1 = 0; y (0) = 2C = 1. So the solution is y = 5 8 sin 2t 1 t cos 2t. 4 11
12 Repeated root canal Yet a worse disease occurs if the LHS has repeated roots and the RHS is a solution of the homogeneous equation. For instance: L[y] = y + 2y + 1 = e t. There is no point trying y = te t because it is also a homogeneous solution. So the next step is to try y = t 2 e t. L kills terms with t 2 or t. The only remaining term is 2e t. So a solution is: y = 1 2 t2 e t. 12
13 What else can the RHS be? So far we have solved L[y] = g when Ly = ay + by + cy and where g is an exponential or a cosine or a sine. We can also solve by undetermined coefficients if the RHS is a polynomial. This is because derivatives of polynomials are polynomials. The degree goes down with each derivative. If the RHS is a polynomial of degree n, and if c 0, you can use a polynomial of degree n. E.g.: L[y] = y + y = t Try y = at 2 + bt + c. Then L[at 2 + bt + c] = 2a + at 2 + bt + c = t a + c = 2, b = 0, a = 1. 13
14 Another example Example: y 3y = 4t 2 1. We can t use a polynomial of degree 2 since the LHS will then be of degree one. Try y = polynomial of degree 3. Thus, y = At 3 +Bt 2 + Ct. We now have 3 undetermined coefficients (A, B, C). Plugging in, we get: (6At + 2B) 3(3At 2 + 2tB + C) = 4t 2 1. Match coefficients of like powers of t: 9A = 4, 6A 6B = 0, 3C = 1. 14
15 What else can the RHS be? A nice thing is that we can easily solve when g = g 1 + g 2 if we can solve separately for g 1 and for g 2. Indeed, if L[Y 1 ] = g 1, L[Y 2 ] = g 2 = L[Y 1 + Y 2 ] = g 1 + g 2 = g. This is the superposition principle again (i.e. linearity of L). So when L = ad 2 + bd + c, our inventory of g s now includes: sums of exponentials, sines, cosines and polynomials. 15
16 What else can the RHS be? If we can solve when the RHS is an exponential, then we can surely solve when the exponential is a complex one. That means we can solve if the RHS g = e λt cos t or g = e λt sin t or a sum of these. One can also solve when the RHS is a polynomial times an exponential. 16
17 Examples L[y] = y 4y = 2t + e 2t. We solve one term at a time. First, we find y 1 such that L[y 1 ] = 2t. We can use at + b. Then L[y] = 4at 4b and we get a = 1 2, b = 0. Then we find y 2 such that L[y 2 ] = e 2t. This is a solution of the homogeneous problem so we must use Ate 2t. We get L[Ate 2t ] = 4Ae 2t = e 2t = A = 1 4. The solution is: y = 1 2 t te2t. 17
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard
More information9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients
September 29, 201 9-1 9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients We have seen that in order to find the general solution to the second order differential
More informationLecture Notes for Math250: Ordinary Differential Equations
Lecture Notes for Math250: Ordinary Differential Equations Wen Shen 2011 NB! These notes are used by myself. They are provided to students as a supplement to the textbook. They can not substitute the textbook.
More informationNonhomogeneous Linear Equations
Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where
More informationMath 22B, Homework #8 1. y 5y + 6y = 2e t
Math 22B, Homework #8 3.7 Problem # We find a particular olution of the ODE y 5y + 6y 2e t uing the method of variation of parameter and then verify the olution uing the method of undetermined coefficient.
More informationGeneral Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More information3.2 Sources, Sinks, Saddles, and Spirals
3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients
More information19.6. Finding a Particular Integral. Introduction. Prerequisites. Learning Outcomes. Learning Style
Finding a Particular Integral 19.6 Introduction We stated in Block 19.5 that the general solution of an inhomogeneous equation is the sum of the complementary function and a particular integral. We have
More informationHW6 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) November 14, 2013. Checklist: Section 7.8: 1c, 2, 7, 10, [16]
HW6 Solutions MATH D Fall 3 Prof: Sun Hui TA: Zezhou Zhang David November 4, 3 Checklist: Section 7.8: c,, 7,, [6] Section 7.9:, 3, 7, 9 Section 7.8 In Problems 7.8. thru 4: a Draw a direction field and
More informationA First Course in Elementary Differential Equations. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A First Course in Elementary Differential Equations Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction Field of y = f(t, y)
More informationSecond-Order Linear Differential Equations
Second-Order Linear Differential Equations A second-order linear differential equation has the form 1 Px d 2 y dx 2 dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. We saw in Section 7.1
More informationCOMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS
COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS BORIS HASSELBLATT CONTENTS. Introduction. Why complex numbers were introduced 3. Complex numbers, Euler s formula 3 4. Homogeneous differential equations 8 5.
More informationSystem of First Order Differential Equations
CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions
More informationMethods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College
Methods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College Equations of Order One: Mdx + Ndy = 0 1. Separate variables. 2. M, N homogeneous of same degree:
More informationHow To Solve A Linear Dierential Equation
Dierential Equations (part 2): Linear Dierential Equations (by Evan Dummit, 2012, v. 1.00) Contents 4 Linear Dierential Equations 1 4.1 Terminology.................................................. 1 4.2
More informationHigher Order Equations
Higher Order Equations We briefly consider how what we have done with order two equations generalizes to higher order linear equations. Fortunately, the generalization is very straightforward: 1. Theory.
More informationSecond Order Linear Differential Equations
CHAPTER 2 Second Order Linear Differential Equations 2.. Homogeneous Equations A differential equation is a relation involving variables x y y y. A solution is a function f x such that the substitution
More informationPartial Fractions: Undetermined Coefficients
1. Introduction Partial Fractions: Undetermined Coefficients Not every F(s) we encounter is in the Laplace table. Partial fractions is a method for re-writing F(s) in a form suitable for the use of the
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
More informationDiscrete Mathematics: Homework 7 solution. Due: 2011.6.03
EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that
More informationStudent name: Earlham College. Fall 2011 December 15, 2011
Student name: Earlham College MATH 320: Differential Equations Final exam - In class part Fall 2011 December 15, 2011 Instructions: This is a regular closed-book test, and is to be taken without the use
More informationMath 2280 - Assignment 6
Math 2280 - Assignment 6 Dylan Zwick Spring 2014 Section 3.8-1, 3, 5, 8, 13 Section 4.1-1, 2, 13, 15, 22 Section 4.2-1, 10, 19, 28 1 Section 3.8 - Endpoint Problems and Eigenvalues 3.8.1 For the eigenvalue
More informationA Brief Review of Elementary Ordinary Differential Equations
1 A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on
More informationCollege of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions
College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationVieta s Formulas and the Identity Theorem
Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion
More informationr (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t)
Solutions HW 9.4.2 Write the given system in matrix form x = Ax + f r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + We write this as ( ) r (t) θ (t) = ( ) ( ) 2 r(t) θ(t) + ( ) sin(t) 9.4.4 Write the given system
More informationDifferential Equations and Linear Algebra Lecture Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University
Differential Equations and Linear Algebra Lecture Notes Simon J.A. Malham Department of Mathematics, Heriot-Watt University Contents Chapter. Linear second order ODEs 5.. Newton s second law 5.2. Springs
More information2 Integrating Both Sides
2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation
More informationFind all of the real numbers x that satisfy the algebraic equation:
Appendix C: Factoring Algebraic Expressions Factoring algebraic equations is the reverse of expanding algebraic expressions discussed in Appendix B. Factoring algebraic equations can be a great help when
More informationComplex Eigenvalues. 1 Complex Eigenvalues
Complex Eigenvalues Today we consider how to deal with complex eigenvalues in a linear homogeneous system of first der equations We will also look back briefly at how what we have done with systems recapitulates
More informationSECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS
SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A second-order linear differential equation has the form 1 Px d y dx dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise
More informationMath 267 - Practice exam 2 - solutions
C Roettger, Fall 13 Math 267 - Practice exam 2 - solutions Problem 1 A solution of 10% perchlorate in water flows at a rate of 8 L/min into a tank holding 200L pure water. The solution is kept well stirred
More informationPRE-CALCULUS GRADE 12
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationExact Values of the Sine and Cosine Functions in Increments of 3 degrees
Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationMultiplier-accelerator Models on Time Scales
International Journal of Statistics and Economics; [Formerly known as the Bulletin of Statistics & Economics ISSN 0973-7022)]; ISSN 0975-556X,; Spring 2010, Volume 4, Number S10; Copyright 2010 by CESER
More informationENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE ORDINARY DIFFERENTIAL EQUATIONS
Texas College Mathematics Journal Volume 6, Number 2, Pages 18 24 S applied for(xx)0000-0 Article electronically published on September 23, 2009 ENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationLectures 5-6: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More informationSection 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.
. The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and
More informationtegrals as General & Particular Solutions
tegrals as General & Particular Solutions dy dx = f(x) General Solution: y(x) = f(x) dx + C Particular Solution: dy dx = f(x), y(x 0) = y 0 Examples: 1) dy dx = (x 2)2 ;y(2) = 1; 2) dy ;y(0) = 0; 3) dx
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationPOLYNOMIALS and FACTORING
POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use
More informationREVIEW EXERCISES DAVID J LOWRY
REVIEW EXERCISES DAVID J LOWRY Contents 1. Introduction 1 2. Elementary Functions 1 2.1. Factoring and Solving Quadratics 1 2.2. Polynomial Inequalities 3 2.3. Rational Functions 4 2.4. Exponentials and
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More informationRecursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1
Recursion Slides by Christopher M Bourke Instructor: Berthe Y Choueiry Fall 007 Computer Science & Engineering 35 Introduction to Discrete Mathematics Sections 71-7 of Rosen cse35@cseunledu Recursive Algorithms
More informationSolutions to Linear First Order ODE s
First Order Linear Equations In the previous session we learned that a first order linear inhomogeneous ODE for the unknown function x = x(t), has the standard form x + p(t)x = q(t) () (To be precise we
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More informationIntroduction to Complex Fourier Series
Introduction to Complex Fourier Series Nathan Pflueger 1 December 2014 Fourier series come in two flavors. What we have studied so far are called real Fourier series: these decompose a given periodic function
More informationSolving Cubic Polynomials
Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial
More information1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More informationMath 2400 - Numerical Analysis Homework #2 Solutions
Math 24 - Numerical Analysis Homework #2 Solutions 1. Implement a bisection root finding method. Your program should accept two points, a tolerance limit and a function for input. It should then output
More informationECG590I Asset Pricing. Lecture 2: Present Value 1
ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1$ now or 1$ one year from now, then you would rather have your money now. If you have to decide
More informationIn the above, the number 19 is an example of a number because its only positive factors are one and itself.
Math 100 Greatest Common Factor and Factoring by Grouping (Review) Factoring Definition: A factor is a number, variable, monomial, or polynomial which is multiplied by another number, variable, monomial,
More informationHow To Factor By Grouping
Lecture Notes Factoring by the AC-method page 1 Sample Problems 1. Completely factor each of the following. a) 4a 2 mn 15abm 2 6abmn + 10a 2 m 2 c) 162a + 162b 2ax 4 2bx 4 e) 3a 2 5a 2 b) a 2 x 3 b 2 x
More information= C + I + G + NX ECON 302. Lecture 4: Aggregate Expenditures/Keynesian Model: Equilibrium in the Goods Market/Loanable Funds Market
Intermediate Macroeconomics Lecture 4: Introduction to the Goods Market Review of the Aggregate Expenditures model and the Keynesian Cross ECON 302 Professor Yamin Ahmad Components of Aggregate Demand
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationLINES AND PLANES CHRIS JOHNSON
LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3-space, as well as define the angle between two non-parallel planes, and determine the distance
More information(Refer Slide Time: 01:11-01:27)
Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 6 Digital systems (contd.); inverse systems, stability, FIR and IIR,
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationSecond Order Linear Partial Differential Equations. Part I
Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; - point boundary value problems; Eigenvalues and Eigenfunctions Introduction
More informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationMATH 31B: MIDTERM 1 REVIEW. 1. Inverses. yx 3y = 1. x = 1 + 3y y 4( 1) + 32 = 1
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inverses. Let f() = 3. Find the inverse g() for f. Solution: Setting y = ( 3) and solving for gives and g() = +3. y 3y = = + 3y y. Let f() = 4 + 3. Find a domain on
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationChapter 2 Solving Linear Programs
Chapter 2 Solving Linear Programs Companion slides of Applied Mathematical Programming by Bradley, Hax, and Magnanti (Addison-Wesley, 1977) prepared by José Fernando Oliveira Maria Antónia Carravilla A
More informationFrequency Response of FIR Filters
Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input
More informationName: ID: Discussion Section:
Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: This exam consists of 6 questions: 4 multiple choice questions worth 5 points each 2 hand-graded questions worth a total of 30 points. INSTRUCTIONS:
More informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
More information12.4 UNDRIVEN, PARALLEL RLC CIRCUIT*
+ v C C R L - v i L FIGURE 12.24 The parallel second-order RLC circuit shown in Figure 2.14a. 12.4 UNDRIVEN, PARALLEL RLC CIRCUIT* We will now analyze the undriven parallel RLC circuit shown in Figure
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationSIGNAL PROCESSING & SIMULATION NEWSLETTER
1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty
More information(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its
(1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:
More informationFactoring Polynomials
Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationSection 1.7 22 Continued
Section 1.5 23 A homogeneous equation is always consistent. TRUE - The trivial solution is always a solution. The equation Ax = 0 gives an explicit descriptions of its solution set. FALSE - The equation
More informationRoots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian)
Roots, Linear Factors, and Sign Charts review of background material for Math 16A (Barsamian) Contents 1. Introduction 1. Roots 1. Linear Factors 4. Sign Charts 5 5. Eercises 8 1. Introduction The sign
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationTMA4213/4215 Matematikk 4M/N Vår 2013
Norges teknisk naturvitenskapelige universitet Institutt for matematiske fag TMA43/45 Matematikk 4M/N Vår 3 Løsningsforslag Øving a) The Fourier series of the signal is f(x) =.4 cos ( 4 L x) +cos ( 5 L
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationHow do we obtain the solution, if we are given F (t)? First we note that suppose someone did give us one solution of this equation
1 Green s functions The harmonic oscillator equation is This has the solution mẍ + kx = 0 (1) x = A sin(ωt) + B cos(ωt), ω = k m where A, B are arbitrary constants reflecting the fact that we have two
More informationJ.L. Kirtley Jr. Electric network theory deals with two primitive quantities, which we will refer to as: 1. Potential (or voltage), and
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.061 Introduction to Power Systems Class Notes Chapter 1: eiew of Network Theory J.L. Kirtley Jr. 1 Introduction
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationElementary Functions
Chapter Three Elementary Functions 31 Introduction Complex functions are, of course, quite easy to come by they are simply ordered pairs of real-valued functions of two variables We have, however, already
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationPaper II ( CALCULUS ) Shahada. College, Navapur. College, Shahada. Nandurbar
Paper II ( CALCULUS ) Prof. R. B. Patel Dr. B. R. Ahirrao Prof. S. M. Patil Prof. A. S. Patil Prof. G. S. Patil Prof. A. D. Borse Art, Science & Comm. College, Shahada Jaihind College, Dhule Art, Science
More information