To determine To verify: The given functions are solutions of the differential equation and determine their Wronskian. Explanation Given: The differential equation x 3 y ‴ + x 2 y ″ − 2 x y ′ + 2 y = 0 and the functions x , x 2 and 1 x . Theorem used: “If the functions p 1 , p 2 , … , p n , and g are continuous on the open interval I , then there exist exactly on solution y = ϕ ( t ) of the differential equation L [ y ] = d n y d t n + p 1 ( t ) d n − 1 y d t n − 1 + ⋯ + p n − 1 ( t ) d y d t + p n ( t ) y = g ( t ) that also satisfies the initial conditions y ( t 0 ) = y 0 , y ′ ( t 0 ) = y ′ 0 , … , y n − 1 ( t 0 ) = y 0 ( n − 1 ) , where t 0 is any point in I . this solution exists throughout the interval I .” Calculation: Consider the function y 1 = x , y 2 = x 2 and y 3 = 1 x . Now, verify that y 1 = x is the solution of the equation x 3 y ‴ + x 2 y ″ − 2 x y ′ + 2 y = 0 . Differentiate y 1 = x with respect to t . y ′ 1 = 1 Differentiate y ′ 1 = 1 with respect to t . y 1 ′ ′ = 0 Differentiate y 1 ′ ′ = 0 with respect to t . y 1 ′ ′ ′ = 0 Now substitute y 1 ′ ′ ′ , y 1 ′ ′ , y 1 ′ and y 1 in x 3 y ‴ + x 2 y ″ − 2 x y ′ + 2 y = 0 . x 3 y ‴ + x 2 y ″ − 2 x y ′ + 2 y = x 3 ( 0 ) + x 2 ( 0 ) − 2 x ( 1 ) + 2 ( x ) = − 2 x + 2 x = 0 Hence, for y 1 = x the differential equation get satisfied. Now, verify that y 2 = x 2 is the solution of the equation x 3 y ‴ + x 2 y ″ − 2 x y ′ + 2 y = 0 . Differentiate y 2 = x 2 with respect to t . y ′ 2 = 2 x Differentiate y ′ 2 = 2 x with respect to t . y 2 ′ ′ = 2 Differentiate y 2 ′ ′ = 2 with respect to t . y 2 ′ ′ ′ = 0 Now substitute y 2 ′ ′ ′ , y 2 ′ ′ , y 2 ′ and y 2 in x 3 y ‴ + x 2 y ″ − 2 x y ′ + 2 y = 0 . x 3 y ‴ + x 2 y ″ − 2 x y ′ + 2 y = x 3 ( 0 ) + x 2 ( 2 ) − 2 x ( 2 x ) + 2 ( x 2 ) = 2 x 2 − 4 x 2 + 2 x 2 = 0 Hence, for y 2 = x 2 the differential equation get satisfied

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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 4.1, Problem 16P

To determine

To verify: The given functions are solutions of the differential equation and determine their Wronskian.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.1, Problem 15P

Chapter 4.1, Problem 17P

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.

Recommended textbooks for you

Advanced Engineering Mathematics
Advanced Math
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:9780073397924
Author:Steven C. Chapra Dr., Raymond P. Canale
Publisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat...
Advanced Math
ISBN:9781118141809
Author:Nathan Klingbeil
Publisher:WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,
Basic Technical Mathematics
Advanced Math
ISBN:9780134437705
Author:Washington
Publisher:PEARSON
Topology
Advanced Math
ISBN:9780134689517
Author:Munkres, James R.
Publisher:Pearson,

Advanced Engineering Mathematics

Advanced Math

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:9780073397924

Author:Steven C. Chapra Dr., Raymond P. Canale

Publisher:McGraw-Hill Education

Introductory Mathematics for Engineering Applicat...

Advanced Math

ISBN:9781118141809

Author:Nathan Klingbeil

Publisher:WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:9781337798310

Author:Peterson, John.

Publisher:Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:9780134437705

Author:Washington

Publisher:PEARSON

Topology

Advanced Math

ISBN:9780134689517

Author:Munkres, James R.

Publisher:Pearson,

SEE MORE TEXTBOOKS

01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY

Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY

Solution of Differential Equations and Initial Value Problems; Author: Jefril Amboy;https://www.youtube.com/watch?v=Q68sk7XS-dc;License: Standard YouTube License, CC-BY