Explanation Consider boundary value problem as follows: x 2 y ″ + x y ' + 9 λ y = 0 ...... ( 1 ) Assume x = e z and simplify the substitution as follows: x = e z log x = z ( Take logarithm both sides ) 1 x d x d y = d z d y ( differentiate both sides with respect to y ) d y d z = x d y d x Further, simplify the above equation as follows: d 2 y d z 2 = d d z ( x d y d x ) ( differentiate both sides with respect to z ) d 2 y d z 2 = x d d z ( d y d x ) + d y d x d d z ( x ) d 2 y d z 2 = x d d z ( d y d x ) + d y d x ( x ) ( d x d z = x ) d 2 y d z 2 = x ( x d 2 y d x 2 ) + d y d x ( x ) ( d d z ( d y d x ) = x d 2 y d x 2 ) Further, simplify the above equation as follows: d 2 y d z 2 = x 2 d 2 y d x 2 + x d y d x d 2 y d z 2 = x 2 y ″ + x y ′ So, equation (1) can be written as follows: d 2 y d z 2 + 9 λ y = 0 ( d 2 y d z 2 = x 2 y ″ + x y ′ ) ...... ( 2 ) Write the Auxiliary equation of equation (2) as follows: m 2 + 9 λ = 0 m 2 = − 9 λ When λ = 0 or λ is negative, then the obtain solution is trivial as below: y ″ = 0 , λ = 0 ...... ( 3 ) The general solution of the differential equation (2) is as follows: y ( z ) = c 1 + c 2 z

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

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 11, Problem 19RE

To determine

The Eigen value and Eigen function of boundary value problem x2y″+xy'+9λy=0, y'(1)=0,y(e)=0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11, Problem 18RE

Chapter 11, Problem 20RE

Students have asked these similar questions

Analyze the residuals of a linear regression model and select the best response.Criteria is simple evaluation of possible indications of an exponential model vs. linear model) no, the residual plot does not show a curve yes, the residual plot does not show a curve yes, the residual plot shows a curve no, the residual plot shows a curve I selected: yes, the residual plot shows a curve and it is INCORRECT. Can u help me understand why?

You have been hired as an intern to run analyses on the data and report the results back to Sarah; the five questions that Sarah needs you to address are given below. please do it step by step on excel Does there appear to be a positive or negative relationship between price and screen size? Use a scatter plot to examine the relationship. Determine and interpret the correlation coefficient between the two variables. In your interpretation, discuss the direction of the relationship (positive, negative, or zero relationship). Also discuss the strength of the relationship. Estimate the relationship between screen size and price using a simple linear regression model and interpret the estimated coefficients. (In your interpretation, tell the dollar amount by which price will change for each unit of increase in screen size). Include the manufacturer dummy variable (Samsung=1, 0 otherwise) and estimate the relationship between screen size, price and manufacturer dummy as a multiple…

(a) (b) (c) (d) de unique? Answer the following questions related to the linear system x + y + z = 2 x-y+z=0 2x + y 2 3 rewrite the linear system into the matrix-vector form A = 5 Fuse elementary row operation to solve this linear system. Is the solution use elementary row operation to find the inverse of A and then solve the linear system. Verify the solution is the same as (b). give the null space of matrix A and find the dimension of null space. give the column space of matrix A and find the dimension of the column space of A (Hint: use Rank-Nullity Theorem).

Chapter 11 Solutions

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

Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...

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 Eigen value and Eigen function of boundary value problem x 2 y ″ + x y ' + 9 λ y = 0 , y ' ( 1 ) = 0 , y ( e ) = 0 .

Solution Summary: The author explains the Eigen value and Eigen function of boundary value problem x2y′′+xy

Videos

The Eigen value and Eigen function of boundary value problem x2y″+xy'+9λy=0, y'(1)=0,y(e)=0.

Want to see the full answer?

Chapter 11 Solutions