Homework #6 Solutions

Size: px
Start display at page:

Download "Homework #6 Solutions"

Transcription

1 Problems Section.1: 6, 4, 40, 46 Section.:, 8, 10, 14, 18, 4, 0 Homework #6 Solutions.1.6. Determine whether the functions f (x) = cos x + sin x and g(x) = cos x sin x are linearly dependent or linearly independent on the real line. Solution: We compute the Wronskian of these functions: W( f, g) = f g f g = cos x + sin x cos x sin x sin x + cos x sin x cos x = ( cos x + sin x)( sin x cos x) ( cos x sin x)( sin x + cos x) = ( 1 cos x sin x 9 sin x 4 cos x) + (1 cos x sin x 9 cos x 4 sin x) = 1(sin x + cos x) = 1 Since the Wronskian is the constant function 1, which is not the 0 function, these functions are linearly independent on the real line (and in fact on any subinterval of the real line) Find the general solution of the DE y + y 15y = 0. Solution: Guessing the solution y = e rx, we obtain the characteristic equation r r + 15 = 0, which factors as (r 5)(r + ) = 0. Therefore, r = 5 and r = are roots, so y 1 = e 5x and y = e x are solutions. Furthermore, by Theorem 5 in.1, the general solution to the DE is y = c 1 e 5x + c e x Find the general solution of the DE 9y 1y + 4y = 0. Solution: As above, the characteristic equation for the DE is 9r 1r + 4 = 0, which factors as (r ) = 0. Therefore, this equation has a double root at r = /. By Theorem 6 in.1, the general solution is then y = c 1 xe x/ + c e x/. 1

2 Find a homogeneous second-order DE ay + by + cy = 0 with general solution y = c 1 e 10x + c e 100x. Solution: This constant coefficient DE must have y 1 = e 10x and y = e 100x as solutions, so we expect r 10 and r 100 to be factors of its characteristic polynomial. Then we may take ar + br + c = a(r 10)(r 100) = a(r 110r ), so taking a = 1, we have the corresponding DE y 110y y = Show directly that the functions f (x) = 5, g(x) = x, and h(x) = x are linearly dependent on the real line. Solution: We find a nontrivial linear combination c 1 f + c g + c h of these functions identically equal to 0. Since all functions are polynomials in x, the function is 0 exactly when the coefficients on all the powers of x are 0. Since c 1 f + c g + c h = c 1 (5) + c ( x ) + c ( x ) = (5c 1 + c + 10c ) + ( c + 15c )x, we require that 5c 1 + c + 10c = 0 and c + 15c = 0. From the second equation, c = 5c. Substituting this into the first, 5c 1 + c + 10c = 5c 1 + (5c ) + 10c = 5c 1 + 0c = 0. Then c 1 = 4c, and there are no more constraints on the c i. Choosing to set c = 1, c 1 = 4 and c = 5. We check that this nontrivial linear combination of functions is 0: ( 4)(5) + (5)( x ) + (1)( x ) = x x = 0. Rearranging this equation, we can express any single one of these functions as a linear combination of the other two: for example, x = 4(5) 5(10 x )...8. Use the Wronskian to prove that the functions f (x) = e x, g(x) = e x, and h(x) = e x are linearly independent on the real line. Solution: We compute W( f, g, h): f g h f g h f g h = e x e x e x e x e x e x e x 4e x 9e x ( = e x e x e x 4e x 9e x e x e x 4e x 9e x + e x e x e x e x ) = e x e x e x (( 9 4 ) ( ) + (1 1 )) = e 6x ( ) = e 6x. Since W(x) = e 6x, which is not zero on the real line (and in fact nowhere 0), these three functions are linearly independent.

3 ..10. Use the Wronskian to prove that the functions f (x) = e x, g(x) = x, and h(x) = x ln x are linearly independent on the interval x > 0. Solution: We compute W( f, g, h). First, we compute derivatives of h: h (x) = x ln x + x 1 = (1 ln x)x x h (x) = ( )(1 ln x)x 4 + x x = (6 ln x 5)x 4 Plugging these into the Wronskian, we have f g h f g h f g h = e x x x ln x e x x (1 ln x)x e x 6x 4 (6 ln x 5)x 4. Rather than expand this directly, we make use of some additional properties of the determinant. One of these is that the determinant is unchanged if a multiple of one column is added to or subtracted from a different column. We subtract ln x times the second column from the third to cancel the ln x terms there: e x x 0 e x x x e x 6x 4 5x 4 Using another property of the determinant, we factor the scalar e x out of the first column, so that it multiplies the determinant of the remaining matrix: e x 1 x 0 1 x x 1 6x 4 5x 4 With these simplifications, we expand along the first row, which conveniently contains a 0 entry: ( e x x x 6x 4 5x 4 1 x ) x 1 5x ( ) = e x 10x 7 6x 7 x ( 5x 4 x ) = e x x 7 (x + 5x + 4) = ex (x + 1)(x + 4) x 7. This function is defined and continuous for all x > 0. Furthermore, none of the factors in its numerator is 0 for x > 0, so it is in fact nowhere 0 on this interval. Since their Wronskian is not identically 0, these functions are linearly independent on this interval.

4 ..14. Find a particular solution to the DE y () 6y + 11y 6y = 0 matching the initial conditions y(0) = 0, y (0) = 0, y (0) = that is a linear combination of y 1 = e x, y = e x, and y = e x. Solution: We let y = c 1 e x + c e x + c e x. Then y = c 1 e x + c e x + c e x and y = c 1 e x + 4c e x + 9c e x, so evaluating these functions at x = 0 and matching them to the initial conditions, we obtain the linear system c 1 + c + c = 0 c 1 + c + c = 0 c 1 + 4c + 9c = We solve this linear system by row reduction of an augmented matrix to echelon form: R R R 1, R R R R R R R 1 R, R R R, R 1 R 1 R R 1 R 1 R Then c 1 = c = and c =, so y = ex e x + ex is the solution to the IVP Find a particular solution to the DE y () y + 4y y = 0 matching the initial conditions y(0) = 1, y (0) = 0, y (0) = 0 that is a linear combination of y 1 = e x, y = e x cos x, and y = e x sin x. Solution: We let y = c 1 e x + c e x cos x + c e x sin x. Then y = c 1 e x + c e x (cos x sin x) + c e x (sin x + cos x) y = c 1 e x c e x sin x + c e x cos x Evaluating these functions at x = 0 and matching them to the initial conditions, we obtain the linear system c 1 + c = 1 c 1 + c + c = 0 c 1 + c = 0 4

5 By the third equation, c 1 = c. Substituting this into the second, c c = 0, so c = c. Finally, in the first equation, c + c = 1, so c = 1, c = 1, and c 1 = ( 1) =. Then y = e x e x cos x e x sin x = e x ( cos x sin x) is the solution to the IVP...4. The nonhomogeneous DE y y + y = x has the particular solution y p = x + 1 and the complementary solution y c = c 1 e x cos x + c e x sin x. Find a solution satisfying the initial conditions y(0) = 4, y (0) = 8. Solution: The general solution to this DE is of the form Then y = y p + y c = x c 1 e x cos x + c e x sin x. y = 1 + c 1 e x (cos x sin x) + c e x (sin x + cos x). Evaluating at x = 0 and applying the initial conditions, y(0) = 1 + c 1 = 4 and y (0) = 1 + c 1 + c = 8. Then c 1 = and c = 4, so the solution to the IVP is y = x e x ( cos x + 4 sin x)...0. Verify that y 1 = x and y = x are linearly independent solutions on the entire real line of the equation x y xy + y = 0, but that W(x, x ) vanishes at x = 0. Why do these observations not contradict part (b) of Theorem? Solution: We first check that these are solutions: x y 1 xy 1 + y 1 = x (0) x(1) + (x) = x( + ) = 0 x y xy + y = x () x(x) + (x ) = x ( 4 + ) = 0 We then compute their Wronskian: W(x, x ) = x x 1 x = x(x) 1(x ) = x. This function is not identically 0, so the two functions x and x are linearly independent on the real line, but it is 0 at precisely x = 0. We note that Theorem applies only to normalized homogeneous linear DEs. Normalizing this DE, we obtain y x y + x y = 0, the coefficient functions of which are continuous for x = 0. Thus, any interval on which the theorem applies does not include x = 0, the only point at which W(x) = 0, so W(x, x ) is nonzero on every such interval. This is consistent with the linear independence of the solutions x and x. 5

Determine whether the following system has a trivial solution or non-trivial solution:

Determine whether the following system has a trivial solution or non-trivial solution: Practice Questions Lecture # 7 and 8 Question # Determine whether the following system has a trivial solution or non-trivial solution: x x + x x x x x The coefficient matrix is / R, R R R+ R The corresponding

More information

MATH 3321 Sample Questions for Exam 3. 3y y, C = Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA

MATH 3321 Sample Questions for Exam 3. 3y y, C = Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA MATH 33 Sample Questions for Exam 3. Find x and y so that x 4 3 5x 3y + y = 5 5. x = 3/7, y = 49/7. Let A = 3 4, B = 3 5, C = 3 Perform the indicated operations, if possible: a AC b AB c B + AC d CBA AB

More information

LINEAR DIFFERENTIAL EQUATIONS. Theorem 1 (Existence and Uniqueness). [1, NSS, Section 6.1, Theorem 1] 1 Suppose. y(x)

LINEAR DIFFERENTIAL EQUATIONS. Theorem 1 (Existence and Uniqueness). [1, NSS, Section 6.1, Theorem 1] 1 Suppose. y(x) LINEAR DIFFERENTIAL EQUATIONS MINSEON SHIN 1. Existence and Uniqueness Theorem 1 (Existence and Uniqueness). [1, NSS, Section 6.1, Theorem 1] 1 Suppose p 1 (x),..., p n (x) and g(x) are continuous real-valued

More information

Chapter 13: General Solutions to Homogeneous Linear Differential Equations

Chapter 13: General Solutions to Homogeneous Linear Differential Equations Worked Solutions 1 Chapter 13: General Solutions to Homogeneous Linear Differential Equations 13.2 a. Verifying that {y 1, y 2 } is a fundamental solution set: We have y 1 (x) = cos(2x) y 1 (x) = 2 sin(2x)

More information

Math 240 Calculus III

Math 240 Calculus III DE Higher Order Calculus III Summer 2015, Session II Tuesday, July 28, 2015 Agenda DE 1. of order n An example 2. constant-coefficient linear Introduction DE We now turn our attention to solving linear

More information

17.2 Nonhomogeneous Linear Equations. 27 September 2007

17.2 Nonhomogeneous Linear Equations. 27 September 2007 17.2 Nonhomogeneous Linear Equations 27 September 2007 Nonhomogeneous Linear Equations The differential equation to be studied is of the form ay (x) + by (x) + cy(x) = G(x) (1) where a 0, b, c are given

More information

Math 2142 Homework 5 Part 1 Solutions

Math 2142 Homework 5 Part 1 Solutions Math 2142 Homework 5 Part 1 Solutions Problem 1. For the following homogeneous second order differential equations, give the general solution and the particular solution satisfying the given initial conditions.

More information

Additional Homework Problems

Additional Homework Problems Math 216 2016-2017 Fall Additional Homework Problems 1 In parts (a) and (b) assume that the given system is consistent For each system determine all possibilities for the numbers r and n r where r is the

More information

Theory of Higher-Order Linear Differential Equations

Theory of Higher-Order Linear Differential Equations Chapter 6 Theory of Higher-Order Linear Differential Equations 6.1 Basic Theory A linear differential equation of order n has the form a n (x)y (n) (x) + a n 1 (x)y (n 1) (x) + + a 0 (x)y(x) = b(x), (6.1.1)

More information

CHAPTER 2. Techniques for Solving. Second Order Linear. Homogeneous ODE s

CHAPTER 2. Techniques for Solving. Second Order Linear. Homogeneous ODE s A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES A COLLECTION OF HANDOUTS ON SCALAR LINEAR ORDINARY DIFFERENTIAL

More information

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1

More information

A Brief Review of Elementary Ordinary Differential Equations

A Brief Review of Elementary Ordinary Differential Equations A 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

Linear Independence x

Linear Independence x Linear Independence A consistent system of linear equations with matrix equation Ax = b, where A is an m n matrix, has a solution set whose graph in R n is a linear object, that is, has one of only n +

More information

Chapter 3. Reading assignment: In this chapter we will cover Sections dx 1 + a 0(x)y(x) = g(x). (1)

Chapter 3. Reading assignment: In this chapter we will cover Sections dx 1 + a 0(x)y(x) = g(x). (1) Chapter 3 3 Introduction Reading assignment: In this chapter we will cover Sections 3.1 3.6. 3.1 Theory of Linear Equations Recall that an nth order Linear ODE is an equation that can be written in the

More information

Midterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015

Midterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Midterm 1 Review Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Summary This Midterm Review contains notes on sections 1.1 1.5 and 1.7 in your

More information

Lecture 18: Section 4.3

Lecture 18: Section 4.3 Lecture 18: Section 4.3 Shuanglin Shao November 6, 2013 Linear Independence and Linear Dependence. We will discuss linear independence of vectors in a vector space. Definition. If S = {v 1, v 2,, v r }

More information

MATH 2331 Linear Algebra. Section 1.1 Systems of Linear Equations. Finding the solution to a set of two equations in two variables: Example 1: Solve:

MATH 2331 Linear Algebra. Section 1.1 Systems of Linear Equations. Finding the solution to a set of two equations in two variables: Example 1: Solve: MATH 2331 Linear Algebra Section 1.1 Systems of Linear Equations Finding the solution to a set of two equations in two variables: Example 1: Solve: x x = 3 1 2 2x + 4x = 12 1 2 Geometric meaning: Do these

More information

Math 240 Calculus III

Math 240 Calculus III Calculus III Summer 2015, Session II Monday, August 3, 2015 Agenda 1. 2. Introduction The reduction of technique, which applies to second- linear differential equations, allows us to go beyond equations

More information

0.1 Problems to solve

0.1 Problems to solve 0.1 Problems to solve Homework Set No. NEEP 547 Due September 0, 013 DLH Nonlinear Eqs. reducible to first order: 1. 5pts) Find the general solution to the differential equation: y = [ 1 + y ) ] 3/. 5pts)

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

Chapter 1: Systems of Linear Equations

Chapter 1: Systems of Linear Equations Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where

More information

Test #2 Math 2250 Summer 2003

Test #2 Math 2250 Summer 2003 Test #2 Math 225 Summer 23 Name: Score: There are six problems on the front and back of the pages. Each subpart is worth 5 points. Show all of your work where appropriate for full credit. ) Show the following

More information

Assignment # 7, Math 370, Fall 2018 SOLUTIONS: y = e x. y = e 3x + 4xe 3x. y = e x cosx.

Assignment # 7, Math 370, Fall 2018 SOLUTIONS: y = e x. y = e 3x + 4xe 3x. y = e x cosx. Assignment # 7, Math 370, Fall 2018 SOLUTIONS: Problem 1: Solve the equations (a) y 8y + 7y = 0, (i) y(0) = 1, y (0) = 1. Characteristic equation: α 2 8α+7 = 0, = 64 28 = 36 so α 1,2 = (8 ±6)/2 and α 1

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian.

MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian. MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian. Spanning set Let S be a subset of a vector space V. Definition. The span of the set S is the smallest subspace W V that contains S. If

More information

3. Identify and find the general solution of each of the following first order differential equations.

3. Identify and find the general solution of each of the following first order differential equations. Final Exam MATH 33, Sample Questions. Fall 7. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C x is the general solution of a differential

More information

Chapter 4. Higher-Order Differential Equations

Chapter 4. Higher-Order Differential Equations Chapter 4 Higher-Order Differential Equations i THEOREM 4.1.1 (Existence of a Unique Solution) Let a n (x), a n,, a, a 0 (x) and g(x) be continuous on an interval I and let a n (x) 0 for every x in this

More information

Exam 2 Solutions. (a) Is W closed under addition? Why or why not? W is not closed under addition. For example,

Exam 2 Solutions. (a) Is W closed under addition? Why or why not? W is not closed under addition. For example, Exam 2 Solutions. Let V be the set of pairs of real numbers (x, y). Define the following operations on V : (x, y) (x, y ) = (x + x, xx + yy ) r (x, y) = (rx, y) Check if V together with and satisfy properties

More information

Worksheet # 2: Higher Order Linear ODEs (SOLUTIONS)

Worksheet # 2: Higher Order Linear ODEs (SOLUTIONS) Name: November 8, 011 Worksheet # : Higher Order Linear ODEs (SOLUTIONS) 1. A set of n-functions f 1, f,..., f n are linearly independent on an interval I if the only way that c 1 f 1 (t) + c f (t) +...

More information

Lecture 22: Section 4.7

Lecture 22: Section 4.7 Lecture 22: Section 47 Shuanglin Shao December 2, 213 Row Space, Column Space, and Null Space Definition For an m n, a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn, the vectors r 1 = [ a 11 a 12 a 1n

More information

Find the solution set of 2x 3y = 5. Answer: We solve for x = (5 + 3y)/2. Hence the solution space consists of all vectors of the form

Find the solution set of 2x 3y = 5. Answer: We solve for x = (5 + 3y)/2. Hence the solution space consists of all vectors of the form Math 2 Homework #7 March 4, 2 7.3.3. Find the solution set of 2x 3y = 5. Answer: We solve for x = (5 + 3y/2. Hence the solution space consists of all vectors of the form ( ( ( ( x (5 + 3y/2 5/2 3/2 x =

More information

Linear Algebra Practice Problems

Linear Algebra Practice Problems Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a

More information

Ordinary Differential Equations (ODEs)

Ordinary Differential Equations (ODEs) Chapter 13 Ordinary Differential Equations (ODEs) We briefly review how to solve some of the most standard ODEs. 13.1 First Order Equations 13.1.1 Separable Equations A first-order ordinary differential

More information

UNDETERMINED COEFFICIENTS SUPERPOSITION APPROACH *

UNDETERMINED COEFFICIENTS SUPERPOSITION APPROACH * 4.4 UNDETERMINED COEFFICIENTS SUPERPOSITION APPROACH 19 Discussion Problems 59. Two roots of a cubic auxiliary equation with real coeffi cients are m 1 1 and m i. What is the corresponding homogeneous

More information

6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if. (a) v 1,, v k span V and

6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if. (a) v 1,, v k span V and 6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if (a) v 1,, v k span V and (b) v 1,, v k are linearly independent. HMHsueh 1 Natural Basis

More information

Series Solutions of Differential Equations

Series Solutions of Differential Equations Chapter 6 Series Solutions of Differential Equations In this chapter we consider methods for solving differential equations using power series. Sequences and infinite series are also involved in this treatment.

More information

6 Second Order Linear Differential Equations

6 Second Order Linear Differential Equations 6 Second Order Linear Differential Equations A differential equation for an unknown function y = f(x) that depends on a variable x is any equation that ties together functions of x with y and its derivatives.

More information

Basic Theory of Linear Differential Equations

Basic Theory of Linear Differential Equations Basic Theory of Linear Differential Equations Picard-Lindelöf Existence-Uniqueness Vector nth Order Theorem Second Order Linear Theorem Higher Order Linear Theorem Homogeneous Structure Recipe for Constant-Coefficient

More information

Chapter 4: Higher Order Linear Equations

Chapter 4: Higher Order Linear Equations Chapter 4: Higher Order Linear Equations MATH 351 California State University, Northridge April 7, 2014 MATH 351 (Differential Equations) Ch 4 April 7, 2014 1 / 11 Sec. 4.1: General Theory of nth Order

More information

Matrices and RRE Form

Matrices and RRE Form Matrices and RRE Form Notation R is the real numbers, C is the complex numbers (we will only consider complex numbers towards the end of the course) is read as an element of For instance, x R means that

More information

Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination

Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination Winfried Just, Ohio University September 22, 2017 Review: The coefficient matrix Consider a system of m linear equations in n variables.

More information

Lecture 6: Spanning Set & Linear Independency

Lecture 6: Spanning Set & Linear Independency Lecture 6: Elif Tan Ankara University Elif Tan (Ankara University) Lecture 6 / 0 Definition (Linear Combination) Let v, v 2,..., v k be vectors in (V,, ) a vector space. A vector v V is called a linear

More information

Linear Algebra 1 Exam 2 Solutions 7/14/3

Linear Algebra 1 Exam 2 Solutions 7/14/3 Linear Algebra 1 Exam Solutions 7/14/3 Question 1 The line L has the symmetric equation: x 1 = y + 3 The line M has the parametric equation: = z 4. [x, y, z] = [ 4, 10, 5] + s[10, 7, ]. The line N is perpendicular

More information

CHAPTER 5. Higher Order Linear ODE'S

CHAPTER 5. Higher Order Linear ODE'S A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 2 A COLLECTION OF HANDOUTS ON SCALAR LINEAR ORDINARY

More information

and let s calculate the image of some vectors under the transformation T.

and let s calculate the image of some vectors under the transformation T. Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

More information

Linear DifferentiaL Equation

Linear DifferentiaL Equation Linear DifferentiaL Equation Massoud Malek The set F of all complex-valued functions is known to be a vector space of infinite dimension. Solutions to any linear differential equations, form a subspace

More information

Review for Exam 2. Review for Exam 2.

Review for Exam 2. Review for Exam 2. Review for Exam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Exam covers: Regular-singular points (5.5). Euler differential equation

More information

MATH 2050 Assignment 6 Fall 2018 Due: Thursday, November 1. x + y + 2z = 2 x + y + z = c 4x + 2z = 2

MATH 2050 Assignment 6 Fall 2018 Due: Thursday, November 1. x + y + 2z = 2 x + y + z = c 4x + 2z = 2 MATH 5 Assignment 6 Fall 8 Due: Thursday, November [5]. For what value of c does have a solution? Is it unique? x + y + z = x + y + z = c 4x + z = Writing the system as an augmented matrix, we have c R

More information

3. Identify and find the general solution of each of the following first order differential equations.

3. Identify and find the general solution of each of the following first order differential equations. Final Exam MATH 33, Sample Questions. Fall 6. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C is the general solution of a differential

More information

Work sheet / Things to know. Chapter 3

Work sheet / Things to know. Chapter 3 MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients

More information

MA 262, Fall 2017, Final Version 01(Green)

MA 262, Fall 2017, Final Version 01(Green) INSTRUCTIONS MA 262, Fall 2017, Final Version 01(Green) (1) Switch off your phone upon entering the exam room. (2) Do not open the exam booklet until you are instructed to do so. (3) Before you open the

More information

1. Why don t we have to worry about absolute values in the general form for first order differential equations with constant coefficients?

1. Why don t we have to worry about absolute values in the general form for first order differential equations with constant coefficients? 1. Why don t we have to worry about absolute values in the general form for first order differential equations with constant coefficients? Let y = ay b with y(0) = y 0 We can solve this as follows y =

More information

Vector Spaces 4.4 Spanning and Independence

Vector Spaces 4.4 Spanning and Independence Vector Spaces 4.4 and Independence Summer 2017 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set

More information

Math 369 Exam #2 Practice Problem Solutions

Math 369 Exam #2 Practice Problem Solutions Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.

More information

EXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1)

EXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1) EXERCISE SET 5. 6. The pair (, 2) is in the set but the pair ( )(, 2) = (, 2) is not because the first component is negative; hence Axiom 6 fails. Axiom 5 also fails. 8. Axioms, 2, 3, 6, 9, and are easily

More information

Key Point. The nth order linear homogeneous equation with constant coefficients

Key Point. The nth order linear homogeneous equation with constant coefficients General Solutions of Higher-Order Linear Equations In section 3.1, we saw the following fact: Key Point. The nth order linear homogeneous equation with constant coefficients a n y (n) +... + a 2 y + a

More information

كلية العلوم قسم الرياضيات المعادالت التفاضلية العادية

كلية العلوم قسم الرياضيات المعادالت التفاضلية العادية الجامعة اإلسالمية كلية العلوم غزة قسم الرياضيات المعادالت التفاضلية العادية Elementary differential equations and boundary value problems المحاضرون أ.د. رائد صالحة د. فاتن أبو شوقة 1 3 4 5 6 بسم هللا

More information

374: Solving Homogeneous Linear Differential Equations with Constant Coefficients

374: Solving Homogeneous Linear Differential Equations with Constant Coefficients 374: Solving Homogeneous Linear Differential Equations with Constant Coefficients Enter this Clear command to make sure all variables we may use are cleared out of memory. In[1]:= Clearx, y, c1, c2, c3,

More information

The Theory of Second Order Linear Differential Equations 1 Michael C. Sullivan Math Department Southern Illinois University

The Theory of Second Order Linear Differential Equations 1 Michael C. Sullivan Math Department Southern Illinois University The Theory of Second Order Linear Differential Equations 1 Michael C. Sullivan Math Department Southern Illinois University These notes are intended as a supplement to section 3.2 of the textbook Elementary

More information

Elementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test Find the radius of convergence of the power series

Elementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test Find the radius of convergence of the power series Elementary Differential Equations, Section 2 Prof. Loftin: Practice Test Problems for Test 2 SOLUTIONS 1. Find the radius of convergence of the power series Show your work. x + x2 2 + x3 3 + x4 4 + + xn

More information

Midterm 1 NAME: QUESTION 1 / 10 QUESTION 2 / 10 QUESTION 3 / 10 QUESTION 4 / 10 QUESTION 5 / 10 QUESTION 6 / 10 QUESTION 7 / 10 QUESTION 8 / 10

Midterm 1 NAME: QUESTION 1 / 10 QUESTION 2 / 10 QUESTION 3 / 10 QUESTION 4 / 10 QUESTION 5 / 10 QUESTION 6 / 10 QUESTION 7 / 10 QUESTION 8 / 10 Midterm 1 NAME: RULES: You will be given the entire period (1PM-3:10PM) to complete the test. You can use one 3x5 notecard for formulas. There are no calculators nor those fancy cellular phones nor groupwork

More information

AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 4. BASES AND DIMENSION

AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 4. BASES AND DIMENSION 4. BASES AND DIMENSION Definition Let u 1,..., u n be n vectors in V. The vectors u 1,..., u n are linearly independent if the only linear combination of them equal to the zero vector has only zero scalars;

More information

Second Order Differential Equations Lecture 6

Second Order Differential Equations Lecture 6 Second Order Differential Equations Lecture 6 Dibyajyoti Deb 6.1. Outline of Lecture Repeated Roots; Reduction of Order Nonhomogeneous Equations; Method of Undetermined Coefficients Variation of Parameters

More information

Chapter 3 Higher Order Linear ODEs

Chapter 3 Higher Order Linear ODEs Chapter 3 Higher Order Linear ODEs Advanced Engineering Mathematics Wei-Ta Chu National Chung Cheng University wtchu@cs.ccu.edu.tw 1 2 3.1 Homogeneous Linear ODEs 3 Homogeneous Linear ODEs An ODE is of

More information

LINEAR EQUATIONS OF HIGHER ORDER. EXAMPLES. General framework

LINEAR EQUATIONS OF HIGHER ORDER. EXAMPLES. General framework Differential Equations Grinshpan LINEAR EQUATIONS OF HIGHER ORDER. EXAMPLES. We consider linear ODE of order n: General framework (1) x (n) (t) + P n 1 (t)x (n 1) (t) + + P 1 (t)x (t) + P 0 (t)x(t) = 0

More information

Lecture 6 & 7. Shuanglin Shao. September 16th and 18th, 2013

Lecture 6 & 7. Shuanglin Shao. September 16th and 18th, 2013 Lecture 6 & 7 Shuanglin Shao September 16th and 18th, 2013 1 Elementary matrices 2 Equivalence Theorem 3 A method of inverting matrices Def An n n matrice is called an elementary matrix if it can be obtained

More information

1 Determinants. 1.1 Determinant

1 Determinants. 1.1 Determinant 1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to

More information

MATH 304 Linear Algebra Lecture 20: Review for Test 1.

MATH 304 Linear Algebra Lecture 20: Review for Test 1. MATH 304 Linear Algebra Lecture 20: Review for Test 1. Topics for Test 1 Part I: Elementary linear algebra (Leon 1.1 1.4, 2.1 2.2) Systems of linear equations: elementary operations, Gaussian elimination,

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra Linear Equations in Linear Algebra.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v,, v p } in n is said to be linearly independent if the vector equation x x x 2 2 p

More information

The Corrected Trial Solution in the Method of Undetermined Coefficients

The Corrected Trial Solution in the Method of Undetermined Coefficients Definition of Related Atoms The Basic Trial Solution Method Symbols Superposition Annihilator Polynomial for f(x) Annihilator Equation for f(x) The Corrected Trial Solution in the Method of Undetermined

More information

Row Space, Column Space, and Nullspace

Row Space, Column Space, and Nullspace Row Space, Column Space, and Nullspace MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Every matrix has associated with it three vector spaces: row space

More information

Higher Order Linear ODEs

Higher Order Linear ODEs c03.qxd 6/18/11 2:57 PM Page 57 CHAPTER 3 Higher Order Linear ODEs Chapters 1 and 2 demonstrate and illustrate that first- and second-order ODEs are, by far, the most important ones in usual engineering

More information

Linear Independence. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics

Linear Independence. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics Linear Independence MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Given a set of vectors {v 1, v 2,..., v r } and another vector v span{v 1, v 2,...,

More information

Math 215 HW #9 Solutions

Math 215 HW #9 Solutions Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith

More information

Span & Linear Independence (Pop Quiz)

Span & Linear Independence (Pop Quiz) Span & Linear Independence (Pop Quiz). Consider the following vectors: v = 2, v 2 = 4 5, v 3 = 3 2, v 4 = Is the set of vectors S = {v, v 2, v 3, v 4 } linearly independent? Solution: Notice that the number

More information

Higher Order Linear ODEs

Higher Order Linear ODEs im03.qxd 9/21/05 11:04 AM Page 59 CHAPTER 3 Higher Order Linear ODEs This chapter is new. Its material is a rearranged and somewhat extended version of material previously contained in some of the sections

More information

Applications of Linear Higher-Order DEs

Applications of Linear Higher-Order DEs Week #4 : Applications of Linear Higher-Order DEs Goals: Solving Homogeneous DEs with Constant Coefficients - Second and Higher Order Applications - Damped Spring/Mass system Applications - Pendulum 1

More information

1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det

1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix

More information

HW2 Solutions

HW2 Solutions 8.024 HW2 Solutions February 4, 200 Apostol.3 4,7,8 4. Show that for all x and y in real Euclidean space: (x, y) 0 x + y 2 x 2 + y 2 Proof. By definition of the norm and the linearity of the inner product,

More information

EE5120 Linear Algebra: Tutorial 3, July-Dec

EE5120 Linear Algebra: Tutorial 3, July-Dec EE5120 Linear Algebra: Tutorial 3, July-Dec 2017-18 1. Let S 1 and S 2 be two subsets of a vector space V such that S 1 S 2. Say True/False for each of the following. If True, prove it. If False, justify

More information

Problem 1 (Equations with the dependent variable missing) By means of the substitutions. v = dy dt, dv

Problem 1 (Equations with the dependent variable missing) By means of the substitutions. v = dy dt, dv V Problem 1 (Equations with the dependent variable missing) By means of the substitutions v = dy dt, dv dt = d2 y dt 2 solve the following second-order differential equations 1. t 2 d2 y dt + 2tdy 1 =

More information

Math 102, Winter 2009, Homework 7

Math 102, Winter 2009, Homework 7 Math 2, Winter 29, Homework 7 () Find the standard matrix of the linear transformation T : R 3 R 3 obtained by reflection through the plane x + z = followed by a rotation about the positive x-axes by 6

More information

The matrix will only be consistent if the last entry of row three is 0, meaning 2b 3 + b 2 b 1 = 0.

The matrix will only be consistent if the last entry of row three is 0, meaning 2b 3 + b 2 b 1 = 0. ) Find all solutions of the linear system. Express the answer in vector form. x + 2x + x + x 5 = 2 2x 2 + 2x + 2x + x 5 = 8 x + 2x + x + 9x 5 = 2 2 Solution: Reduce the augmented matrix [ 2 2 2 8 ] to

More information

Lecture 1: Review of methods to solve Ordinary Differential Equations

Lecture 1: Review of methods to solve Ordinary Differential Equations Introductory lecture notes on Partial Differential Equations - c Anthony Peirce Not to be copied, used, or revised without explicit written permission from the copyright owner 1 Lecture 1: Review of methods

More information

9 - Matrix Methods for Linear Systems

9 - Matrix Methods for Linear Systems 9 - Matrix Methods for Linear Systems 9.4 Linear Systems in Normal Form Homework: p. 523-526 # ü Introduction Consider a system of n linear differential equations given by x 1 x 2 x n = a 11 HtL x 1 HtL

More information

Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation.

Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation. Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: y y x y2 = 0 : homogeneous equation. x2 v = y dy, y = vx, and x v + x dv dx = v + v2. dx =

More information

Math 54 HW 4 solutions

Math 54 HW 4 solutions Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,

More information

MATH 225 Summer 2005 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 2005

MATH 225 Summer 2005 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 2005 MATH 225 Summer 25 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 25 Department of Mathematical and Statistical Sciences University of Alberta Question 1. [p 224. #2] The set of all

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A x= 0, where

More information

Section 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).

Section 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N). Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results

More information

SECTION 3.3. PROBLEM 22. The null space of a matrix A is: N(A) = {X : AX = 0}. Here are the calculations of AX for X = a,b,c,d, and e. =

SECTION 3.3. PROBLEM 22. The null space of a matrix A is: N(A) = {X : AX = 0}. Here are the calculations of AX for X = a,b,c,d, and e. = SECTION 3.3. PROBLEM. The null space of a matrix A is: N(A) {X : AX }. Here are the calculations of AX for X a,b,c,d, and e. Aa [ ][ ] 3 3 [ ][ ] Ac 3 3 [ ] 3 3 [ ] 4+4 6+6 Ae [ ], Ab [ ][ ] 3 3 3 [ ]

More information

June 2011 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations

June 2011 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations June 20 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations The topics covered in this exam can be found in An introduction to differential equations

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 Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise

More information

LECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS

LECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS LECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS 1. Regular Singular Points During the past few lectures, we have been focusing on second order linear ODEs of the form y + p(x)y + q(x)y = g(x). Particularly,

More information

Math 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016

Math 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016 Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the

More information

Numerical Linear Algebra Homework Assignment - Week 2

Numerical Linear Algebra Homework Assignment - Week 2 Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.

More information

1111: Linear Algebra I

1111: Linear Algebra I 1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Michaelmas Term 2015 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Michaelmas Term 2015 1 / 10 Row expansion of the determinant Our next goal is

More information

Goals: Second-order Linear Equations Linear Independence of Solutions and the Wronskian Homogeneous DEs with Constant Coefficients

Goals: Second-order Linear Equations Linear Independence of Solutions and the Wronskian Homogeneous DEs with Constant Coefficients Week #3 : Higher-Order Homogeneous DEs Goals: Second-order Linear Equations Linear Independence of Solutions and the Wronskian Homogeneous DEs with Constant Coefficients 1 Second-Order Linear Equations

More information

CHAPTER 1 Systems of Linear Equations

CHAPTER 1 Systems of Linear Equations CHAPTER Systems of Linear Equations Section. Introduction to Systems of Linear Equations. Because the equation is in the form a x a y b, it is linear in the variables x and y. 0. Because the equation cannot

More information

Section 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation

Section 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results

More information