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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 443
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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Short Answer

Question: In Problems 1–10, determine all the singular points of the given differential equation.

8. exy"-(x2-1)y'+2xy=0

There is no singularity point that exists in this differential equation for both P(X) and Q(X).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Ordinary and Singular Points

A point X0 is called an ordinary point of equation y"+p(x)y'+q(x)y=0 if both p and q are analytic at X0 . If X0 is not an ordinary point, it is called a singular point of the equation.

Step 2: Find the singular points

The given differential equation is

exy"-(x2-1)y'+2xy=0

Dividing the above equation by ex we get,

y"-(x2-1)exy'+2xexy=0

On comparing the above equation with y"+p(x)y'+q(x)y=0 ,we find that,

Q(x)=2xex

Hence, Px) and Q(x) are analytic except, perhaps, when their denominators are zero.

For Px) this occurs at no point. In this case both Px) and Q(x) are analytic at all points.

Therefore, there is no singularity point that exists in this differential equation for both Px) and Q(x) .

Most popular questions for Math Textbooks

To derive the general solutions given by equations (17)- (20) for the non-homogeneous equation (16), complete the following steps.

(a) Substitute y(x)=∑n=0∞anxn and the Maclaurin series into equation (16) to obtain

(2a2-a0)+∑k=1∞[(k+2)(k+1)ak+2-(k+1)ak]xk=∑n=0∞(-1)n(2n+1)!x2n+1

(b) Equate the coefficients of like powers on both sides of the equation in part (a) and thereby deduce the equations

a2=a02,a3=16+a13,a4=a08,a5=140+a115,a6=a048,a7=195040+a1105

(c) Show that the relations in part (b) yield the general solution to (16) given in equations (17)-(20).

Question: Compute the Taylor series for f(x)= in(1+x2) about x0 = 0. [Hint: Multiply the series for (1+x2)-1 by 2x and integrate.]

In Problems 13-19, find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

y''-e2xy'+(cosx)y=0y(0)=-1, y'(0)=1

Question: In Problems 29–34, determine the Taylor series about the point x0 for the given functions and values of x0.

29. f(x)= cosx , x0 = π

Question: Let

Show that fn(0)=0 for n=0,1,2.... and hence that the Maclaurin series for f(x) is 0+0+0+.... , which converges for all x but is equal to f(x) only when x=0 . This is an example of a function possessing derivatives of all orders (at x0 =0 ), whose Taylor series converges, but the Taylor series (about x0 =0) does not converge to the original function! Consequently, this function is not analytic at x=0.
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