Explanation Given: The differential equation is 4 x 2 y ″ − 4 x y ′ + ( 16 x 2 + 3 ) y = 0 . Formula used: The solution of the differential equation of the form, y ″ + ( 1 − 2 a x ) y ′ + ( b 2 c 2 x 2 c − 2 + a 2 − p 2 c 2 x 2 ) y = 0 , p is not an integer, in terms of Bessel function is given by y = x a [ c 1 J p ( b x c ) + c 2 J − p ( b x c ) ] . Calculation: consider the differential equation. y ″ + ( 1 − 2 a x ) y ′ + ( b 2 c 2 x 2 c − 2 + a 2 − p 2 c 2 x 2 ) y = 0 ...... ( 1 ) Divide the given differential equation by 4 x 2 . y ″ − 1 x y ′ + ( 4 + 3 4 x 2 ) y = 0 ...... ( 2 ) Compare the coefficient of y ′ and y in equation (1) and (2). − 1 x = ( 1 − 2 a x ) and 4 + 3 4 x 2 = ( b 2 c 2 x 2 c − 2 + a 2 − p 2 c 2 x 2 ) Now the results are obtained as follows: 1 − 2 a = − 1 , 2 c − 2 = 0 , b 2 c 2 = 4 and a 2 − p 2 c 2 = 3 4 Further simplify the above equation. a = 1 , c = 1 , b = 2 and p = 1 2 Substitute the values of above constants in the solution as follows

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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 6.4, Problem 26E

To determine

The general solution of the differential equation 4x2y″−4xy′+(16x2+3)y=0 in terms of Bessel functions.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.4, Problem 25E

Chapter 6.4, Problem 27E

Students have asked these similar questions

Do the Laplace Transformation and give the answer in Partial Fractions. Also do the Inverted Laplace Transformation and explain step-by-step.

I need help with this problem and an explanation of the solution for the image described below. (Statistics: Engineering Probabilities)

12. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.508.XP. ASK YOUR TEA Make a substitution to express the integrand as a rational function and then evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) x + 16 dx X Need Help? Read It SUBMIT ANSWER 13. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.512.XP. ASK YOUR TEA Make a substitution to express the integrand as a rational function and then evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) dx 8)(2x + 1) Need Help? Read It SUBMIT ANSWER 14. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.518.XP. Find the area of the region under the given curve from 1 to 5. y = x² +7 6x - x² Need Help? Read It ASK YOUR TEA

Chapter 6 Solutions

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

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 general solution of the differential equation 4 x 2 y ″ − 4 x y ′ + ( 16 x 2 + 3 ) y = 0 in terms of Bessel functions.

Solution Summary: The author explains the general solution of the differential equation 4x2y′′-4x

Concept explainers

Videos

The general solution of the differential equation 4x2y″−4xy′+(16x2+3)y=0 in terms of Bessel functions.

Want to see the full answer?

Chapter 6 Solutions