3. (a) Show that x = 0 is a regular singular point for the Laguerre equation xy" + (1 − x)y' + 2y = 0, and using the Frobenius method, show that both roots of the indicial equation are equal to zero. Show that the corresponding series for the solution y = Σ anx" is a polynomial and find its explicit form. n=0 (b) Give an example of a second order linear differential equation with polynomial coefficients possessing exactly four singular points such that the points x = ±2 are regular singular points and the points x = ±1 are irregular singular points. Justify your example. (c) Justify the statement: If the equation y" + p(x)y' + q(x)y = 0 admits a solution y = x(ex-e-2x), then the point x = 0 cannot be an ordinary point of the equation.
3. (a) Show that x = 0 is a regular singular point for the Laguerre equation xy" + (1 − x)y' + 2y = 0, and using the Frobenius method, show that both roots of the indicial equation are equal to zero. Show that the corresponding series for the solution y = Σ anx" is a polynomial and find its explicit form. n=0 (b) Give an example of a second order linear differential equation with polynomial coefficients possessing exactly four singular points such that the points x = ±2 are regular singular points and the points x = ±1 are irregular singular points. Justify your example. (c) Justify the statement: If the equation y" + p(x)y' + q(x)y = 0 admits a solution y = x(ex-e-2x), then the point x = 0 cannot be an ordinary point of the equation.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Question
![3. (a) Show that x = 0 is a regular singular point for the Laguerre equation
xy" + (1 − x)y' + 2y = 0,
and using the Frobenius method, show that both roots of the indicial equation are
equal to zero. Show that the corresponding series for the solution y = Σ anx" is
a polynomial and find its explicit form.
n=0
(b) Give an example of a second order linear differential equation with polynomial
coefficients possessing exactly four singular points such that the points x = ±2 are
regular singular points and the points x = ±1 are irregular singular points. Justify
your example.
(c) Justify the statement: If the equation y" + p(x)y' + q(x)y = 0 admits a solution
y = x(ex-e-2x), then the point x = 0 cannot be an ordinary point of the equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69cce4ac-4bf6-4e6b-8636-bf160e045b58%2F8481e40a-968a-48be-abe1-f9820188fa84%2F7vpqg3w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. (a) Show that x = 0 is a regular singular point for the Laguerre equation
xy" + (1 − x)y' + 2y = 0,
and using the Frobenius method, show that both roots of the indicial equation are
equal to zero. Show that the corresponding series for the solution y = Σ anx" is
a polynomial and find its explicit form.
n=0
(b) Give an example of a second order linear differential equation with polynomial
coefficients possessing exactly four singular points such that the points x = ±2 are
regular singular points and the points x = ±1 are irregular singular points. Justify
your example.
(c) Justify the statement: If the equation y" + p(x)y' + q(x)y = 0 admits a solution
y = x(ex-e-2x), then the point x = 0 cannot be an ordinary point of the equation.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 3 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)