4. (a) Solve the differential equation (1 − x 2 )y ′′ − 2xy′ + λ(λ + 1)y = 0 using power series centered at 0 , in which λ is a constant. Write your solution as a linear combination of two independent solutions whose coefficients are expressed in terms of λ . Compute the coefficients of each solution up to and including the x 5 term. Without computing them, what is the smallest possible value of the radius of...
4. (a) Solve the differential equation (1 − x 2 )y ′′ − 2xy′ + λ(λ + 1)y = 0 using power series centered at 0 , in which λ is a constant. Write your solution as a linear combination of two independent solutions whose coefficients are expressed in terms of λ . Compute the coefficients of each solution up to and including the x 5 term. Without computing them, what is the smallest possible value of the radius of...
Consider the differential equation (1 2 yay 0, where a E R is a constant. (a) By analysing the equation, show that there are two linearly independent power series solutions in powers of for la<1 (b) Find two linearly independent solutions. Note: The recurrence relation you derive should be the following (or equivalent to it) (n-a)(n a) an (n1)(n2) n 2 0. an+2 polynomial solution of (c) Show that if a is (nonnegative) integer n, then there is a degree...
Question 1 4 pts To find a power series solution about x = 0 to y + 2xy = 0, which are procedures needed? Apply the Theorem 3 that all coefficients must be O to determine the coefficients an Show x = 0 is an ordinary point. Shift the indices so that the general term in each is a constant times ck and combined these power series as only one series. All of them Write the solution as a power...
4. (a) Solve the differential equation (1-12)y"-2cy' + λ(A + 1)y 0 using power series centered at 0 , in which λ is a constant. Write your solution as a linear combination of two independent solutions whose coefficients are expressed in terms of λ . Compute the coefficients of each solution up to and including the 5 term. Without computing them, what is the smallest possible value of the radius of convergence of each solution and why? (b) When λ...
1) Find all the values of x such that the given series would converge. M-1 Inn+ 7) 3) Wrte the Taylor polynomial T5(for the function f(x) e" centered at z 4) Calculate the Taylor polynomi alsT2(r)andT3(x)centered at X=2for 0. f(x) In(z +1) (Ctrl)
1) Find all the values of x such that the given series would converge. M-1 Inn+ 7) 3) Wrte the Taylor polynomial T5(for the function f(x) e" centered at z 4) Calculate the Taylor polynomi alsT2(r)andT3(x)centered at...
10.5.3 Consider the defining differential equation for the Hermite polynomials do and solve it by the series solution method for functions Hn(x such that Hx)exp(-x2/2) can be normalized In your solution (i) find a recurrence relation between the coefficients of the power series solutions [Note: this (ii) show that Hn(x)exp(x/2) wll not be normalizable unless the power series terminates (ii) choosing co 0 or 1 and c0 or 1, find the first 5 power series solutions of the equation. relation...
Number 11 please. And please explain the final step to your y=
equation
10–14 SERIES SOLUTIONS Find a power series solution in powers of x. Show the details. 10. y" - y' + xy = 0 11. y" - y' + x’y = 0 12. (1 - x?)y" - 2xy' + 2y = 0 13. y" + (1 + x2)y = 0 14. y" - 4xy' + (4x2 – 2y = 0 ons
Please help with this question. Thank you!
1. We say p (ro. yo, 20) is a regular point for the equation F(x, y,) 0 if the equation either defines as a differentiable function f( for (, y) in a neighborhood of (ro, Vo), or defines y as a differentiable function y-g(, a) for (r, z) in a neighborhood of (ro, 2o), or defines z as a differentiable functionh(x, y) for (x, y) in a neighborhood of (ro.o). a. Suppose p...
Use a power series centered about the ordinary point x0 = 0 to solve the differential equation (x − 4)y′′ − y′ + 12xy = 0 Find the recurrence relation and at least the first four nonzero terms of each of the two linearly inde- pendent solutions (unless the series terminates sooner). What is the guaranteed radius of convergence?