n==0 converges for all x. ndidsode noia In each of Problems 1 through 4 determine d"(xn), d"(xo), and o (Xo) for = if y (x) is a solution of the given initial value problem. THonbios 1. y" + xy' +y = 0; y(0) = 1, y (0) = 0 2. y"+ (sin x)y + (cos x)y = 0; 3. x²y" + (1+ x)y' +3(ln x)y = 0; 4 y"+x²y' + (sin x)y = 0; y(0) = 0, y'(0) = 1 %3D y(1) = 2, y'(1) = 0 %3D %3D .50° y(0) = ao, y' (0) = a1 %3D In each of Problems 5 through 8 determine a lower bound for the radius o series solutions about each given point xo for the given differential equation is ng Le 5. y" + 4y' + 6xy = 0; 6. (x2-2x - 3)y" +xy' +4y = 0; nioxo 1) (1+x')y" + 4xy' +y = 0; Xo = 0, xo = 4 %3D %3D = 4, xo = -4, xo = 02A night uloz %3D (7, xo = 0, xo =2 %3D Jassl 8. ry" +y = 0; Xo = 1 %3D goituiedua edh god 9, Determine a lower bound for the radius of convergence of series solutions x, for each of the differential equations in Problems 1 through 14 of Sect 10. The Chebyshev Equation. The Chebyshev' differential equation is (1 –x)y"- xy' +a²y = 0, %3D where a is a constant. (a) Determine two linearly independent solutions in powers of x for |xl (b) Show that if a is a nonnegative integer n, then there is a polynor degree n. These polynomials, when properly normalized, are called polynomials. They are very useful in problems that require a polynomia to a function defined on -1 < x < 1. (c) Find a polynomial solution for each of the cases a =n=0,1,2, 3. For each of the differential equations in Problems 11 through 14 find the firs terms in each of two linearly independent power series solutions about th you expect the radius of convergence to be for eaçh soluti 11. y" + (sinru
n==0 converges for all x. ndidsode noia In each of Problems 1 through 4 determine d"(xn), d"(xo), and o (Xo) for = if y (x) is a solution of the given initial value problem. THonbios 1. y" + xy' +y = 0; y(0) = 1, y (0) = 0 2. y"+ (sin x)y + (cos x)y = 0; 3. x²y" + (1+ x)y' +3(ln x)y = 0; 4 y"+x²y' + (sin x)y = 0; y(0) = 0, y'(0) = 1 %3D y(1) = 2, y'(1) = 0 %3D %3D .50° y(0) = ao, y' (0) = a1 %3D In each of Problems 5 through 8 determine a lower bound for the radius o series solutions about each given point xo for the given differential equation is ng Le 5. y" + 4y' + 6xy = 0; 6. (x2-2x - 3)y" +xy' +4y = 0; nioxo 1) (1+x')y" + 4xy' +y = 0; Xo = 0, xo = 4 %3D %3D = 4, xo = -4, xo = 02A night uloz %3D (7, xo = 0, xo =2 %3D Jassl 8. ry" +y = 0; Xo = 1 %3D goituiedua edh god 9, Determine a lower bound for the radius of convergence of series solutions x, for each of the differential equations in Problems 1 through 14 of Sect 10. The Chebyshev Equation. The Chebyshev' differential equation is (1 –x)y"- xy' +a²y = 0, %3D where a is a constant. (a) Determine two linearly independent solutions in powers of x for |xl (b) Show that if a is a nonnegative integer n, then there is a polynor degree n. These polynomials, when properly normalized, are called polynomials. They are very useful in problems that require a polynomia to a function defined on -1 < x < 1. (c) Find a polynomial solution for each of the cases a =n=0,1,2, 3. For each of the differential equations in Problems 11 through 14 find the firs terms in each of two linearly independent power series solutions about th you expect the radius of convergence to be for eaçh soluti 11. y" + (sinru
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Number 5 5.3
Transcribed Image Text:n==0
converges for all x.
ndidsode noia
In each of Problems 1 through 4 determine d"(xn), d"(xo), and o (Xo) for
= if y (x) is a solution of the given initial value problem.
THonbios
1. y" + xy' +y = 0; y(0) = 1, y (0) = 0
2. y"+ (sin x)y + (cos x)y = 0;
3. x²y" + (1+ x)y' +3(ln x)y = 0;
4 y"+x²y' + (sin x)y = 0;
y(0) = 0, y'(0) = 1
%3D
y(1) = 2, y'(1) = 0
%3D
%3D
.50°
y(0) = ao, y' (0) = a1
%3D
In each of Problems 5 through 8 determine a lower bound for the radius o
series solutions about each given point xo for the given differential equation
is
ng Le
5. y" + 4y' + 6xy = 0;
6. (x2-2x - 3)y" +xy' +4y = 0; nioxo
1) (1+x')y" + 4xy' +y = 0;
Xo = 0, xo = 4
%3D
%3D
= 4, xo = -4, xo = 02A
night
uloz
%3D
(7,
xo = 0, xo =2
%3D
Jassl
8. ry" +y = 0;
Xo = 1
%3D
goituiedua edh god
9, Determine a lower bound for the radius of convergence of series solutions
x, for each of the differential equations in Problems 1 through 14 of Sect
10. The Chebyshev Equation. The Chebyshev' differential equation is
(1 –x)y"- xy' +a²y = 0,
%3D
where a is a constant.
(a) Determine two linearly independent solutions in powers of x for |xl
(b) Show that if a is a nonnegative integer n, then there is a polynor
degree n. These polynomials, when properly normalized, are called
polynomials. They are very useful in problems that require a polynomia
to a function defined on -1 < x < 1.
(c) Find a polynomial solution for each of the cases a =n=0,1,2, 3.
For each of the differential equations in Problems 11 through 14 find the firs
terms in each of two linearly independent power series solutions about th
you expect the radius of convergence to be for eaçh soluti
11. y" + (sinru
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