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(20 pts.) The Laguerre differential equation is ry" + (1 - )y' + Ay = 0....
(20 pts.) The Laguerre differential equation is ry" + (1 - )y' + Ay = 0. (a) Show that x = 0 is a regular singular point. (b) Determine the indicial equation, its roots, and the recurrence relation. (c) Find one solution (x > 0). Show that if = m, a positive integer, this solution reduces to a polynomial. When properly normalized, this polynomial is known as the Laguerre polynomial, L. (2).
Check by direct substitution that the series given below is a solution to the Laguerre equation, ...
Check by direct substitution that the series given
below is a solution to the Laguerre equation, which is the
following:
a)The expression to be checked is the
following:
b) By choosing α appropriately, obtain a general
expression for the Laguerre polynomials.
c) Get the first 6 polynomials of Laguerre.
d) Make a graph Ln (x) (Laguerre polynomial of order
n) versus x. Forcing the coefficient of the highest exponent of x
to have a value equal to 1.
ry" (1...
9. Solve the IVP with Cauchy-Euler ODE: xy"txy+4y-0; y(1)-o, y )--3 = 0 , use Variat...
9. Solve the IVP with Cauchy-Euler ODE: xy"txy+4y-0; y(1)-o, y )--3 = 0 , use Variat 0 10. Given that y = GXtar2 is a solution of the Cauchy-Euler ODE x, "+ 2xy-2 Parameters to find the general solution of the non-homogeneous ODE y+2xy-y homogeneoury"rQ&)e-ar)-
3. Consider the following ODE: (1 + 2%)/" - xy + y = 0 (a) Find...
3. Consider the following ODE: (1 + 2%)/" - xy + y = 0 (a) Find the first 3 nonzero terms of the power series expansion (around x = 0) for the general solution. (b) Use the ratio test to determine the radius of convergence of the series. What can you say about the radius of convergence without solving the ODE? (c) Determine the solution that satisfies the initial conditions y(0) = 1 and (0) = 0.
3. a) Classify each ODE by order and linearity: y" – 3xy' + xy = 0...
3. a) Classify each ODE by order and linearity: y" – 3xy' + xy = 0 b) y(4) + 2xy" - x?y' - xy' + sin y = 0 c) 2.5** 12.5x = sint
I need help with question 30d 16. y = 0 (that is, y(x) = 0 for...
I need help with question 30d
16. y = 0 (that is, y(x) = 0 for all x, also written y(x) = 0) is a solution of (2) (not of (1) if (x) • o , called the trivial solution 17. The sum of a solution of (1) and a solution of (2) is a solution of (1). 18. The difference of two solutions of (1) is a solution of (2). 19. If yı is a solution of (1), what...
From Arfken, obtain recurrence relations for Laguerre polynomials as mentioned in the text. By differentiating the generating function in Eq. (13.56) with respect to x and z, we obtain recurrence re...
From Arfken, obtain recurrence relations for Laguerre
polynomials as mentioned in the text.
By differentiating the generating function in Eq. (13.56) with respect to x and z, we obtain recurrence relations for the LaguerTe polynomials as follows. Using the product rule for differentiation we verify the identities ag ag (13.61) g(x, z)= 2 n=0
By differentiating the generating function in Eq. (13.56) with respect to x and z, we obtain recurrence relations for the LaguerTe polynomials as follows. Using the...
Question 3 Consider the ordinary differential equation (ODE) 2xy" + (1 + x)y' + 3y =...
Question 3 Consider the ordinary differential equation (ODE) 2xy" + (1 + x)y' + 3y = 0, in the neighbourhood of the origin. a) Show that x = 0 is a regular singular point of the ODE. (10) b) By seeking an appropriate solution to the ODE, show that G=- (10) i) the roots to the indicial equation of the ODE are 0 and 1/2. [10] ii) the recurrence formula used to determine the power series coefficients, ens when one...
4) xy" + y' – xy = 0,x, = 0 a) What are the points of...
4) xy" + y' – xy = 0,x, = 0 a) What are the points of singularity for each specific problem? b) Does this ODE hold a general power series solution at the specific xO? Justify your answer, if your answer is yes, the proceed as follows: Compute the radius of analyticity and report the corresponding interval, and Identify the recursion formula for the power series coefficient around xo, and write the corresponding solution with at least four non-zero terms...
Xy', + y,-xy = 0, x,-0 Answer the following questions a) What are the points of singularity for e...
xy', + y,-xy = 0, x,-0 Answer the following questions a) What are the points of singularity for each specific problem? b) Does this ODE hold a general power series solution at the specific x0? Justify your answer, i) if your answer is yes, the proceed as follows: Compute the radius of analyticity and report the corresponding interval, and ldentify the recursion formula for the power series coefficient around x0, and write the corresponding solution with at least four non-zero...
(20 pts.) The Laguerre differential equation is ry" + (1 - )y' + Ay = 0. (a) Show that x = 0 is a regular singular point. (b) Determine the indicial equation, its roots, and the recurrence relation. (c) Find one solution (x > 0). Show that if = m, a positive integer, this solution reduces to a polynomial. When properly normalized, this polynomial is known as the Laguerre polynomial, L. (2).
Check by direct substitution that the series given
below is a solution to the Laguerre equation, which is the
following:
a)The expression to be checked is the
following:
b) By choosing α appropriately, obtain a general
expression for the Laguerre polynomials.
c) Get the first 6 polynomials of Laguerre.
d) Make a graph Ln (x) (Laguerre polynomial of order
n) versus x. Forcing the coefficient of the highest exponent of x
to have a value equal to 1.
ry" (1...
9. Solve the IVP with Cauchy-Euler ODE: xy"txy+4y-0; y(1)-o, y )--3 = 0 , use Variat 0 10. Given that y = GXtar2 is a solution of the Cauchy-Euler ODE x, "+ 2xy-2 Parameters to find the general solution of the non-homogeneous ODE y+2xy-y homogeneoury"rQ&)e-ar)-
3. Consider the following ODE: (1 + 2%)/" - xy + y = 0 (a) Find the first 3 nonzero terms of the power series expansion (around x = 0) for the general solution. (b) Use the ratio test to determine the radius of convergence of the series. What can you say about the radius of convergence without solving the ODE? (c) Determine the solution that satisfies the initial conditions y(0) = 1 and (0) = 0.
3. a) Classify each ODE by order and linearity: y" – 3xy' + xy = 0 b) y(4) + 2xy" - x?y' - xy' + sin y = 0 c) 2.5** 12.5x = sint
I need help with question 30d
16. y = 0 (that is, y(x) = 0 for all x, also written y(x) = 0) is a solution of (2) (not of (1) if (x) • o , called the trivial solution 17. The sum of a solution of (1) and a solution of (2) is a solution of (1). 18. The difference of two solutions of (1) is a solution of (2). 19. If yı is a solution of (1), what...
From Arfken, obtain recurrence relations for Laguerre
polynomials as mentioned in the text.
By differentiating the generating function in Eq. (13.56) with respect to x and z, we obtain recurrence relations for the LaguerTe polynomials as follows. Using the product rule for differentiation we verify the identities ag ag (13.61) g(x, z)= 2 n=0
By differentiating the generating function in Eq. (13.56) with respect to x and z, we obtain recurrence relations for the LaguerTe polynomials as follows. Using the...
Question 3 Consider the ordinary differential equation (ODE) 2xy" + (1 + x)y' + 3y = 0, in the neighbourhood of the origin. a) Show that x = 0 is a regular singular point of the ODE. (10) b) By seeking an appropriate solution to the ODE, show that G=- (10) i) the roots to the indicial equation of the ODE are 0 and 1/2. [10] ii) the recurrence formula used to determine the power series coefficients, ens when one...
4) xy" + y' – xy = 0,x, = 0 a) What are the points of singularity for each specific problem? b) Does this ODE hold a general power series solution at the specific xO? Justify your answer, if your answer is yes, the proceed as follows: Compute the radius of analyticity and report the corresponding interval, and Identify the recursion formula for the power series coefficient around xo, and write the corresponding solution with at least four non-zero terms...
xy', + y,-xy = 0, x,-0 Answer the following questions a) What are the points of singularity for each specific problem? b) Does this ODE hold a general power series solution at the specific x0? Justify your answer, i) if your answer is yes, the proceed as follows: Compute the radius of analyticity and report the corresponding interval, and ldentify the recursion formula for the power series coefficient around x0, and write the corresponding solution with at least four non-zero...
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