Explanation Definition used: Linear dependence and independence: “The functions f 1 , f 2 , … , f n are said to be linearly dependent on an interval I if there exists a set of constants k 1 , k 2 , … , k n not all zero, such that k 1 f 1 ( t ) + k 2 f 2 ( t ) + ⋯ + k n f n ( t ) = 0 for all t in I . The functions f 1 , f 2 , … , f n are said to be linearly independent on I if they are not linearly dependent there.” Theorem used: If the function p 1 , p 2 , … , p n are continuous on the open interval I , if the functions y 1 , y 2 , … , y n are solutions of the equation L [ y ] = y ( n ) + p 1 ( t ) y ( n − 1 ) + ⋯ + p n − 1 ( t ) y ′ + p n ( t ) y = 0 and if W ( y 1 , y 2 , … , y n ) ( t ) ≠ 0 for at least one point in I , then every solution of the equation L [ y ] = y ( n ) + p 1 ( t ) y ( n − 1 ) + ⋯ + p n − 1 ( t ) y ′ + p n ( t ) y = 0 can be expressed as a linear combination of the solution y 1 , y 2 , … , y n . And Wronskian of the function is W ( y 1 , y 2 , … , y n ) = | y 1 y 2 ⋯ y n y 1 ′ y 2 ′ ⋯ y n ′ ⋮ ⋮ ⋮ y 1 ( n − 1 ) y 2 ( n − 1 ) ⋯ y n ( n − 1 ) | . Calculation: Consider the function f 1 ( t ) = 2 t − 3 , f 2 ( t ) = t 3 + 1 , f 3 ( t ) = 2 t 2 − t and f 4 ( t ) = t 2 + t + 1 . Wronskian of the above function can be calculated as follows

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Concept explainers

Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, c…

Videos

Question

Chapter 4.1, Problem 10P

To determine

Whether the given functions are linearly dependent or linearly independent, if it is linearly dependent find the linear relation among them.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.1, Problem 9P

Chapter 4.1, Problem 11P

Students have asked these similar questions

5. (a) State the Residue Theorem. Your answer should include all the conditions required for the theorem to hold. (4 marks) (b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the anti-clockwise direction. Evaluate に dz. You must check all of the conditions of any results that you use. (5 marks) (c) Evaluate L You must check all of the conditions of any results that you use. ཙ x sin(Tx) x²+2x+5 da. (11 marks)

3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula for L(y). (1 mark) (b) Let UCC be open. Let f: U→C be continuous. Let y: [a,b] → U be a contour. Suppose there exists a finite real number M such that |f(z)| < M for all z in the image of y. Prove that < ||, f(z)dz| ≤ ML(y). (3 marks) (c) State and prove Liouville's theorem. You may use Cauchy's integral formula without proof. (d) Let R0. Let w € C. Let (10 marks) U = { z Є C : | z − w| < R} . Let f UC be a holomorphic function such that 0 < |ƒ(w)| < |f(z)| for all z Є U. Show, using the local maximum modulus principle, that f is constant. (6 marks)

3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M a simple module? (b) State and prove Schur's Lemma for simple modules. (c) Let AM(K) and M = K" the natural A-module. (i) Show that M is a simple K-module. (ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a is a matrix in the centre of M, (K). [Recall that the centre, Z(M,(K)) == {a Mn(K) | ab M,,(K)}.] = ba for all bЄ (iii) Explain briefly why this means End₁(M) K, assuming that Z(M,,(K))~ K as K-algebras. Is this consistent with Schur's lemma?

Chapter 4 Solutions

Elementary Differential Equations

Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - Prob. 6P Ch. 4.1 - In each of Problems 7 through 10, determine...Ch. 4.1 - Prob. 8P Ch. 4.1 - In each of Problems 7 through 10, determine...Ch. 4.1 - Prob. 10P

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.