Explanation Definition used: Three functions y 1 ( t ) , y 2 ( t ) and y 3 ( t ) are said to be linearly dependent on an interval I , at least one of these functions is a linear combination of the remaining two. Equivalently, y 1 , y 2 and y 3 are linearly dependent on I if there exist constants C 1 , C 2 and C 3 , not all zero, such that C 1 y 1 ( t ) + C 2 y 2 ( t ) + C 3 y 3 ( t ) = 0 for all t in I . Calculation: The given equation is y ″ − 6 y 2 = 0 . Consider the function y = 1 ( c − t ) 2 , where c = 1 , 2 , 3 . The derivative will be, y ′ = − 2 ( c − t ) − 3 and y ″ = 6 ( c − t ) − 4 . Substitute y = ( c − t ) − 2 and y ″ = 6 ( c − t ) − 4 in y ″ − 6 y 2 = 0 , 6 ( c − t ) − 4 − 6 ( ( c − t ) − 2 ) 2 = 0 6 ( c − t ) − 4 − 6 ( c − t ) − 4 = 0 0 = 0 Since the equality is satisfied, y = 1 ( c − t ) 2 is a solution to y ″ − 6 y 2 = 0 . That is, the functions 1 ( 1 − t ) 2 , 1 ( 2 − t ) 2 and 1 ( 3 − t ) 2 are solutions to y ″ − 6 y 2 = 0

9th Edition

ISBN: 9780135963777

Author: Nagle

Publisher: PEARSON CO

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 4.8, Problem 4E

To determine

To show: The solutions 1(1−t)2,1(2−t)2 and 1(3−t)2 to y″−6y2=0 are linearly independent on the interval (−1,1).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.8, Problem 3E

Chapter 4.8, Problem 5E

Students have asked these similar questions

Q/Draw the graph k 3,4, and extract perfect ma ching from it

π sin2x sin3x sin5x sin30x sinx - sin6x - sin10x - sin15x) 2 dx

f(x+T) = f(x) f(x) = sin² (x) + cos² (x) T=?

Chapter 4 Solutions

EP FUND.OF DIFF.EQUATIONS-MYLAB (18 WK)

Ch. 4.1 - Verify that for b = 0 and Fext(t) = 0, equation...Ch. 4.1 - If Fext(t) = 0, equation (3) becomes my + by + ky...Ch. 4.1 - Show that if Fext(t) = 0, m = 1, k = 9, and b = 6,...Ch. 4.1 - Verify that y = sin 3t + 2 cos 3t is a solution to...Ch. 4.1 - Prob. 5E Ch. 4.1 - An external force F(t) = 2 cos 2t is applied to a...Ch. 4.1 - Prob. 7E Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - Prob. 10E

Knowledge Booster

Learn more about

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

The solutions 1 ( 1 − t ) 2 , 1 ( 2 − t ) 2 and 1 ( 3 − t ) 2 to y ″ − 6 y 2 = 0 are linearly independent on the interval ( − 1 , 1 ) .

Solution Summary: The author explains that three functions are linearly independent on the interval (1-t)2, and one is a linear combination of the remaining two.

Concept explainers

Videos

To show: The solutions 1(1−t)2,1(2−t)2 and 1(3−t)2 to y″−6y2=0 are linearly independent on the interval (−1,1).

Want to see the full answer?

Chapter 4 Solutions