So far, we have looked at homogeneous equations

Size: px
Start display at page:

Download "So far, we have looked at homogeneous equations"

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. 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 information

9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients

9. 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 information

Lecture Notes for Math250: Ordinary Differential Equations

Lecture 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 information

Nonhomogeneous Linear Equations

Nonhomogeneous 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 information

Math 22B, Homework #8 1. y 5y + 6y = 2e t

Math 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 information

General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1

General 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 information

3.2 Sources, Sinks, Saddles, and Spirals

3.2 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 information

19.6. Finding a Particular Integral. Introduction. Prerequisites. Learning Outcomes. Learning Style

19.6. 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 information

HW6 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 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 information

A 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 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 information

Second-Order Linear Differential Equations

Second-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 information

COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS

COMPLEX 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 information

System of First Order Differential Equations

System 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 information

Methods 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 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 information

How To Solve A Linear Dierential Equation

How 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 information

Higher Order Equations

Higher 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 information

Second Order Linear Differential Equations

Second 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 information

Partial Fractions: Undetermined Coefficients

Partial 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 information

Integrals of Rational Functions

Integrals 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 information

Discrete Mathematics: Homework 7 solution. Due: 2011.6.03

Discrete 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 information

Student name: Earlham College. Fall 2011 December 15, 2011

Student 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 information

Math 2280 - Assignment 6

Math 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 information

A Brief Review of Elementary Ordinary Differential Equations

A 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 information

College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions

College 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 information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

The 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 information

Vieta s Formulas and the Identity Theorem

Vieta 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 information

r (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)

r (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 information

Differential 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 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 information

2 Integrating Both Sides

2 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 information

Find all of the real numbers x that satisfy the algebraic equation:

Find 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 information

Complex Eigenvalues. 1 Complex Eigenvalues

Complex 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 information

SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS

SECOND-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 information

Math 267 - Practice exam 2 - solutions

Math 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 information

PRE-CALCULUS GRADE 12

PRE-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 information

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

CITY 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 information

Exact 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 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 information

LS.6 Solution Matrices

LS.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 information

Multiplier-accelerator Models on Time Scales

Multiplier-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 information

ENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE ORDINARY DIFFERENTIAL EQUATIONS

ENCOURAGING 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 information

L 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

L 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 information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear 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 information

Lectures 5-6: Taylor Series

Lectures 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

[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 information

Section 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.

Section 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 information

tegrals as General & Particular Solutions

tegrals 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 information

Zeros of Polynomial Functions

Zeros 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 information

POLYNOMIALS and FACTORING

POLYNOMIALS 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 information

REVIEW EXERCISES DAVID J LOWRY

REVIEW 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 information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression 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 information

Recursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1

Recursive 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 information

Solutions to Linear First Order ODE s

Solutions 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 information

ORDINARY DIFFERENTIAL EQUATIONS

ORDINARY 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 information

Algebra Practice Problems for Precalculus and Calculus

Algebra 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 information

Introduction to Complex Fourier Series

Introduction 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 information

Solving Cubic Polynomials

Solving 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 information

1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.

1.(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 information

A vector is a directed line segment used to represent a vector quantity.

A 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 information

2x + 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

2x + 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 information

Factoring Polynomials and Solving Quadratic Equations

Factoring 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 information

Math 2400 - Numerical Analysis Homework #2 Solutions

Math 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 information

ECG590I Asset Pricing. Lecture 2: Present Value 1

ECG590I 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 information

In the above, the number 19 is an example of a number because its only positive factors are one and itself.

In 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 information

How To Factor By Grouping

How 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

= 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 information

Zeros of Polynomial Functions

Zeros 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 information

LINES AND PLANES CHRIS JOHNSON

LINES 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)

(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 information

Lecture 14: Section 3.3

Lecture 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 information

Second Order Linear Partial Differential Equations. Part I

Second 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 information

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South 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 information

MATH 31B: MIDTERM 1 REVIEW. 1. Inverses. yx 3y = 1. x = 1 + 3y y 4( 1) + 32 = 1

MATH 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 information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST 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 information

Chapter 2 Solving Linear Programs

Chapter 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 information

Frequency Response of FIR Filters

Frequency 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 information

Name: ID: Discussion Section:

Name: 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 information

SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS. 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 information

12.4 UNDRIVEN, PARALLEL RLC CIRCUIT*

12.4 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 information

Algebra. 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. 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 information

SIGNAL PROCESSING & SIMULATION NEWSLETTER

SIGNAL 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. 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 information

Factoring Polynomials

Factoring 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 information

Algebra 2 PreAP. Name Period

Algebra 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 information

Partial 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: 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 information

a 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.

a 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 information

Section 1.7 22 Continued

Section 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 information

Roots, 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 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 information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 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 information

TMA4213/4215 Matematikk 4M/N Vår 2013

TMA4213/4215 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 information

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN 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 information

Algebra and Geometry Review (61 topics, no due date)

Algebra 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 information

How 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

How 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 information

J.L. Kirtley Jr. Electric network theory deals with two primitive quantities, which we will refer to as: 1. Potential (or voltage), and

J.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 information

Math 215 HW #6 Solutions

Math 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 information

Core Maths C2. Revision Notes

Core 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 information

Elementary Functions

Elementary 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 information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 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 information

1.3 Algebraic Expressions

1.3 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 information

6.1 Add & Subtract Polynomial Expression & Functions

6.1 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 information

Paper II ( CALCULUS ) Shahada. College, Navapur. College, Shahada. Nandurbar

Paper 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