Explanation The given initial value problem is 4 x 2 y ″ + y = 0 , y ( − 1 ) = 2 , y ′ ( − 1 ) = 4 . The above equation can be rewritten as 4 t 2 d 2 y d t 2 + y = 0 with initial conditions y ( 1 ) = 2 and y ′ ( 1 ) = − 4 . Then the higher derivatives becomes y ′ = m t m − 1 and y ″ = m ( m − 1 ) t m − 2 . Substitute the values of y and y ″ in the equation 4 t 2 d 2 y d t 2 + y = 0 and obtain the auxiliary equation as follows. 4 t 2 d 2 y d t 2 + y = 0 4 t 2 m ( m − 1 ) t m − 2 + t m = 0 4 m ( m − 1 ) t m + t m = 0 ( 4 m ( m − 1 ) + 1 ) t m = 0 Solve the above auxiliary equation and obtain the roots of m as shown below. 4 m ( m − 1 ) + 1 = 0 4 m 2 − 4 m + 1 = 0 ( 2 m − 1 ) 2 = 0 m 1 = 1 2 , m 2 = 1 2 Therefore, the factors of the above equation are m 1 = 1 2 , m 2 = 1 2 . Since the roots are repeated, the general solution is given by y = c 1 t m + c 2 t m ln t . Therefore, y = c 1 t 1 2 + c 2 t 1 2 ln t . Also, y ′ = 1 2 c 1 t − 1 2 + 1 2 c 2 t − 1 2 ln t + c 2 t 1 2 1 t . Apply the initial conditions y ( 1 ) = 2 and y ′ ( 1 ) = − 4 and obtain the values of the constants c 1 and c 2 as shown below

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

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 4.7, Problem 37E

To determine

The solution of the initial value problem 4x2y″+y=0, y(−1)=2, y′(−1)=4 on the interval (−∞,0) using the substitution t=−x.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.7, Problem 36E

Chapter 4.7, Problem 38E

Students have asked these similar questions

Given: Circle J 2 What is the value of y? A. 38 C. 68 B. 50 D. 92

please find the answers for the yellows boxes using the information and the picture below

Consider the function f(x) = x²-1. (a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative. Show all your steps clearly. (b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the graph where x 1 and x-> 1+h (for a small positive value of h, illustrate conceptually). Then, draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the value you found in part (a). (c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in the context of the graph of f(x). How does the rate of change of this function vary at different points?

Chapter 4 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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 solution of the initial value problem 4 x 2 y ″ + y = 0 , y ( − 1 ) = 2 , y ′ ( − 1 ) = 4 on the interval ( − ∞ , 0 ) using the substitution t = − x .

Solution Summary: The author explains the solution of the initial value problem 4x2y′′+y=0 using the substitution t=-x.

Concept explainers

Videos

The solution of the initial value problem 4x2y″+y=0, y(−1)=2, y′(−1)=4 on the interval (−∞,0) using the substitution t=−x.

Want to see the full answer?

Chapter 4 Solutions